Functionals in Set Theory and Arithmetic

Wolfgang Burr

Für Iris,

meine Schwester Almut

und meine Eltern Gertrud und Wieland

The present script represents the author’s doctoral dissertation at the Mathematisch-Naturwissenschaftliche Fakultät of the Westfälische Wilhelms-Universität Münster which has been developed under the supervision of Professor Justus Diller.

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Foreword

A lot of people have, in some way or another, contributed to the development of my thesis. First of all I want to express my deep gratitude to my thesis advisor Professor Dr. Justus Diller, who taught me mathematical logic and awoke my interest for this subject. He has constantly supported me in various ways and has accompanied the development of my thesis with great interest and effort. Numerous fruitful and stimulating discussions were a source of inspiration. During the last years I have been employed by him as a wissenschaftlicher Mitarbeiterat his Institut für mathematische Logik und Grundlagenforschung. I want to thank him for all this. Professor Wolfram Pohlers has consistently contributed to my education and interest in mathematical logic. I have learned a lot in his lectures and seminars and from various discussions with him. I have gratefully accepted his support and advice. From HDoz. Andreas Weiermann I have also learned a lot about mathematical logic, so many thanks to him for all his useful advice. Following his suggestion I started to study functional interpretations of systems of set theory. I also owe a lot to the members of the Institut für mathematische Logik und Grundlagenforschung. Dr. Kai F. Wehmeier’s questions were the starting point of the second part of my thesis and I thank him for many fruitful discussions. The contact with Dr. Arnold Beckmann and Dr. Benjamin Blankertz enriched my research. I’m grateful to Michael Möllerfeld for numerous discussions and in particular for his suggestion to me to prove the interpretation of foundation by a Skolem-process. It is a great pleasure to have Dr. Lev D. Beklemishev as a guest at our institute at the moment. He put several interesting questions to me and I thank him for inspiring conversations. Volker Hartung’s work about the functional interpretation of KPω was the starting point of the first part of my dissertation. Ingo Lepper contributed a lot to the layout of this thesis. All these people and also the present and former students and Martina Pfeifer and Hildegard Brunstering make our institute a pleasant place to work. I wish also to thank all members of the Philosophy Department, Carnegie Mellon University, Pittsburgh PA for their hospitality during my two months’ stay in summer 1997 and for their invitation to give a talk at the Philosophy Department Logic Colloquium. I am especially grateful to Professor Jeremy Avigad, who made these two months in Pittsburgh fruitful and instructive for me. My further thanks are due to Professors Anne S. Troelstra, Albert Visser and Dick de Jongh

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for their interest in my work, helpful discussions and the invitation to give a talk at the Amsterdam-Utrecht Symposium on Intuitionistic Arithmetic and Modal Logic. Finally, I want to thank Professor Michael Rathjen for profound and instructive discussions about constructive set theory. The Studienstiftung des Deutschen Volkes supported my stay in Pittsburgh generously. Professor Justus Diller and Gertrud Burr gave me useful hints concerning my English. On the private side, I am grateful to my love Iris, my sister Almut and my parents Gertrud and Wieland Burr for all they gave me. I dedicate my thesis to them.

Contents

1 Introduction 1

2 Preliminaries 7 2.a Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.b Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.c Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.d Functional interpretation of arithmetic . . . . . . . . . . . . . . . . . . . . . . 12 2.e Arithmetic in all finite types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

I Set Theory 23

3 Towards a functional interpretation of set theory 25 3.a Concepts and aims of functional interpretations . . . . . . . . . . . . . . . . . . 25 3.b Functional interpretations of set theory . . . . . . . . . . . . . . . . . . . . . . 35

4 Functional interpretation of Aczel’s constructive set theory 43 4.a Systems T_{∈} and T_{∈}^{+} of constructive set functionals . . . . . . . . . . . . . . . . 43 4.b Functional interpretation of CZF^{−} and CZF . . . . . . . . . . . . . . . . . . . 54 4.c Constructive set theories in all finite types . . . . . . . . . . . . . . . . . . . . 69 4.d Hereditarily majorizable functionals . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Functional interpretation of Kripke-Platek set theories 83 5.a Kripke-Platek set theories in all finite types . . . . . . . . . . . . . . . . . . . . 83 5.b KPi, KPM, KPω + (Pow) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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II Arithmetic 91

6 On relations between classical and intuitionistic arithmetic 93

7 Fragments of Heyting arithmetic 97 7.a Fragments iΦ′_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.b Fragments iΦ_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.c Fragments iΦ̃_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.d Fragments iΘ_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.e Characterization of provably recursive and provably total functions . . . . . . . . 119

8 iΔ_{0} + Collection principles 121 8.a iΔ_{0} + {Bφ | φ ∈ L_{A}} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.b iΔ_{0} + (Coll) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Chapter 1

Introduction

In 1958, Gödel published his famous article “Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes” in the Swiss Journal Dialectica. There he gives an interpretation of Heyting arithmetic (HA) in a quantifier free system T of primitive recursive functionals of finite type. This interpretation is rich in consequences and applications, it implies, for instance, the consistency of HA relative to T, gives rise to a characterization of the provably recursive functions of HA and can be used to establish Markov’s principle for Σ_{1}- predicates for HA. In combination with the negative translation it furthermore reduces Peano arithmetic (PA) to T. Gödel’s Dialectica interpretation attracted a lot of attention and it turned out to be a powerful tool in proof theory. Consequently, in the following decades it has been extended to a large number of theories, among them (see [AF98], p. 340): Intuitionistic arithmetic with principles of transfinite induction (G. Kreisel), impredicative classical and intuitionistic analysis (G. Kreisel, C. Spector, W. Howard, H. Luckhardt, J. Diller and H. Vogel, J.-Y. Girard), predicative systems of classical analysis (S. Feferman, W. Maass), Peano arithmetic (J. Shoenfield), subsystems of Peano arithmetic (C. Parsons), theories of ordinals (W. Howard, S. Feferman), Heyting arithmetic in all finite types (J. Diller and W. Nahm), bounded arithmetic (S. Cook and A. Urquhard), iterated arithmetical fixed point theories (J. Avigad). Furthermore, the Dialectica interpretation has been used to obtain various results in proof theory, e.g. the no-counterexample interpretation for PA (G. Kreisel), closure and conservation results for intuitionistic systems (A. S. Troelstra) and furthermore, conservation results for classical systems and characterizations of provably recursive functions (C. Parsons, S. Feferman, U. Kohlenbach).

In the present thesis we continue this line of research. In the first part we extend the Dialectica interpretation to set theory. We give functional interpretations of various systems of set theory, among them Aczel’s constructive set theory CZF, Kripke-Platek set theory with infinity (KPω) and classical and intuitionistic set theories in all finite types.

The second part of the dissertation deals with intuitionistic first-order arithmetic, and here the Dialectica interpretation was the starting point of our work and an important tool to obtain the results. We define a series of fragments iΦ_{n}, iΦ′_{n}, iΘ_{n} of Heyting arithmetic and prove the Π_{2}-equivalence to the well-investigated fragments IΠ_{n} of PA. The classes Φ_{n} turn out to be natural counterparts of the fragments Π_{n} of the arithmetical hierarchy. Furthermore, we investigate iΔ_{0} adjoint by collection principles. We prove that collection is rather weak in an intuitionistic context. On the other side, we show that the contrapositives of the collection axioms give rise to the law of excluded middle.

The results of part I of this dissertation were presented at the Amsterdam-Münster contact (Münster 1998), at the PTAC ’98 Workshop on Proof Theory and Complexity (Aarhus 1998) and at the Colloquium Logicum 98 (Berlin 1998). The results of this part have been submitted for publication. The results of the second part were presented at the Philosophy Department Logic Colloquium (Carnegie Mellon University, Pittsburgh 1997) and at the Amsterdam-Utrecht Symposium on Intuitionistic Arithmetic and Modal Logic (Amsterdam 1997). A prepublication “Fragments of Heyting arithmetic” has been accepted for publication by the Journal of Symbolic Logic.

Extended Summary

The present thesis consists of two parts. The first one deals with functional interpretations of set theory, and the second one with intuitionistic first order arithmetic. Both parts of the thesis can be read independently. The thesis is almost self-contained; in chapter 2 we briefly summarize the definitions and results to which we refer in the sequel. Hence, we suggest that the reader may skip this chapter and refers back to it only if necessary. In the extended summary we cite only the main theorems of the thesis.

Chapter 3 is an introduction to the concepts and aims of functional interpretations with a focus on the functional interpretation of set theory.

In chapter 4 we give functional interpretations of Aczel’s constructive set theory CZF and related systems. CZF is formulated in the language L_{∈} of set theory. Its logic is intuitionistic first order predicate logic. The non-logical axioms of CZF^{−}are (Extensionality), (Pair), (Union), (Infinity), (Foundation), (Δ_{0}-Separation) and (Strong-Collection). CZF in addition contains (Subset-Collection).

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In order to give a functional interpretation of Aczel’s systems, in section 4.a we introduce the set theoretic counterpart of Gödel’s system T or, to be more precise, the counterpart of the variant T_{∧} introduced by Diller and Nahm. That is, we introduce systems T_{∈} and T_{∈}^{+} of constructive set functionals. These systems extend Jensen and Carps primitive recursive set functions to finite types. Basic functionals of T_{∈} are 0 and ω for the corresponding sets, combinators K and S, a successor functional Suc, functionals N and I for relative complement and intersection, a union functional U and finally functionals R for transfinite recursion. The extension T_{∈}^{+} in addition contains a fullness functional. T_{∈} is interesting in its own right and has various nice properties: we can define λ-operators, all usual set operations like union, intersection, etc., separation-terms for Δ_{0}-formulae, restriction functionals, casedistinction functionals, pairing, projections, simultaneous recursion, the transitive closure of a set, ω-recursion etc.

The heart of the first part is section 4.b where we define the translation^{×} for the language L_{∈} of set theory. This translation is a refinement of the Diller-Nahm translation which is again an extension of the Dialectica translation. The new translation allows us to carry through a functional interpretation of constructive set theory. The crucial point is that for every φ ∈ L_{∈} the translation φ^{×} reads

∃vQ∀w[(∃q^{∈}Q)(q = q) ∧ (∀q^{∈}Q)φ̄(vq, w)].

That means the existential quantifiers now range over inhabited sets {vq | q^{∈}Q} where each element vq (q^{∈}Q) interprets the matrix φ̄. For a detailed discussion of this translation and the need for it we refer to chapter 3. The main theorem of 4.b is

Theorem 1 CZF^{−}is ^{×}-interpretable in T_{∈}. CZF is ^{×}-interpretable in T_{∈}^{+}.

The interpretation theorem yields characterizations of the definable set functions of CZF^{−} and CZF in terms of constructive set functionals.

In section 4.c we define constructive set theories in all finite types in order to characterize the translation ^{×}. The natural framework for this characterization is CZF^{ω−}: it is the system ‘spanned’ by T_{∈} and CZF^{−}. For the characterization, we introduce the following schemes:

(WAC) ∀x∃yφ(x, y) → ∃Y S ∀x[(∃s^{∈}Sx)(s = s) ∧ (∀s^{∈}Sx)φ(x, Y sx)]

and further

(IP_{∈}) (∀wφ → ∃uψ) → ∃US[∀wφ → (∃s^{∈}Sx)(s = s) ∧ (∀s^{∈}Sx)ψ(Us)],

where φ ∈ T_{∈} and

(M_{∈}) (∀wφ → ψ) → ∃W X[(∀x^{∈}X)φ(W x) → ψ] φ, ψ ∈ T_{∈}.

Furthermore, the theory CZF_{∗}^{ω−} is defined by adding these three principles to CZF^{ω−}. Our characterization theorem now states:

Theorem 2 CZF_{∗}^{ω−} = CZF^{ω−} + {φ ↔ φ^{×} | φ ∈ L(CZF^{ω−})}.

Furthermore we have

Theorem 3 CZF_{∗}^{ω−} is ^{×}-interpretable in T_{∈}.

This theorem again yields a relative consistency result and gives rise to characterizations of the definable set functions of CZF_{∗}^{ω−}. The previous two theorems also hold if we consider the extensions with (Subset-Collection) and add the fullness functional to T_{∈}.

The aim of section 4.d is to prove that the first non-trivial axiom of extensionality for finite types, namely

(E2) ∀Y αβ(∀u[αu = βu] → Y α = Y β) (Y : 2; α, β : 1)

is not ^{×}-interpretable by functionals of T_{∈}^{+}. The proof is an adaption of Howard’s proof of the non-interpretability of this axiom in the context of arithmetic by hereditarily majorizable functionals. We introduce the notion of hereditarily majorizable set functional and show that all functionals of T_{∈}^{+} are hereditarily majorizable, but none of the majorizable functionals interprets (E2). Hence we obtain

Theorem 4 (E2) is not ^{×}-interpretable in T_{∈}^{+}.

In chapter 5 we introduce and study the Shoenfield variant ^{∨} of the translation ^{×}. With this translation we interpret Kripke-Platek set theory with infinity (KPω) in the classical counterpart T_{∈}^{c} of T_{∈}. Furthermore, we introduce Kripke-Platek set theory in all finite types (KP ω^{ω}) and characterize the translation ^{∨}. Again the interpretation can be extended to KPω^{ω} +{φ ↔ φ^{∨}}. Finally we give extensions of T_{∈}^{c} that are able to interpret KPi, KPM and KPω + (Pow).

Chapter 6 is an introduction to the second part of the thesis.

Chapter 7 deals with fragments of Heyting arithmetic. For Γ a class of formulae iΓ denotes Heyting arithmetic with induction restricted to the formulae of Γ. Similarly, IΓ denotes the classical counterpart. The canonical and well-investigated fragments of PA are IΠ_{n} (or IΣ_{n}, both coincide). The family (IΠ_{n})_{n∈ω} forms a proper hierarchy of PA and the union of all IΠ_{n} is PA. This is due to the prenex normal-form theorem for classical predicate logic: every formula is equivalent to a formula in some Π_{n}. Of course, the prenex normalform theorem does not hold for intuitionistic logic, i.e., there are formulae that are not equivalent to any prenex formula. Thus the fragments iΠ_{n} do not a priori form a hierarchy that exhausts HA. Indeed, prenex induction within an intuitionistic framework is remarkably weak. Visser proved (we quote a sharpened version by Wehmeier [Weh96]): iΔ_{0}+ prenex induction (and hence each iΠ_{n+2}) is Π_{2}-conservative over iΠ_{2}. Consequently, for n > 2 we do not have reasonable intuitionistic counterparts of IΠ_{n} so far. This leads to the subject of chapter 7, where we define classes Φ_{n} such that the following holds:

Theorem 5 (i) IΠ_{n} = IΦ_{n}

(ii) ⋃_{n∈ω} Φ_{n} = L_{A} (modulo equivalence in iΔ_{0})

(iii) IΠ_{n} and iΦ_{n} prove the same Π_{2}-formulae.

So, each fragment iΦ_{n} is a natural counterpart of IΠ_{n}. The definition of Φ_{n} provides a normal-form for formulae of arithmetic in the intuitionistic framework.

Furthermore, we define variants Φ′_{n} of Φ_{n} which contain only almost negative formulae. We show that iΦ′_{n} and IΠ_{n} are still Π_{2}-equivalent. Hence we only need induction for almost negative formulae to obtain the Π_{2}-strength of HA. The definition of Φ′_{n} provides a normal form for almost negative formulae. Furthermore, we define truth-predicates for the formulae classes Φ_{n} and Φ′_{n}. This implies that all fragments iΦ_{n}, iΦ′_{n} are finitely axiomatizable.

In section 7.d we generalize the result by Visser and Wehmeier on prenex induction within HA. Our result is that after closing Φ_{n} both under existential and universal quantification (we call these classes Θ_{n}), the corresponding theories iΘ_{n} still prove the same Π_{2}-formulae. The main tool to establish this result is a Dialectica-interpretation of iΘ_{n} in a suitable fragment of Gödel’s T.

In section 7.e we investigate provably total functions, i.e., functions that are provably total and definable by arbitrary formulae. Classical predicate logic already proves the totality of a non-recursive function. In contrast, for intuitionistic theories it often happens that the class of provably recursive and provably total functions coincide. We show in in Section 7.e that this also holds for our fragments:

Theorem 6 Let S be either iΘ_{n+1} or iΦ_{n+1}. Suppose S ⊢ ∀x∃yφ(x, y) with φ ∈ L_{A} arbitrary. Then there is a ≺ ω_{n+2}-descent-recursive function f such that HA^{ω∗} ⊢ φ(x, fx).

In chapter 8 we consider iΔ_{0} plus collection-principles. The collection schema Bφ for a formula φ is

Bφ :≡ ∀x ⩽ a∃yφ(x, y) → ∃z∀x ⩽ a∃y ⩽ zφ(x, y).

It is well known that IΔ_{0} + {Bφ | φ ∈ L_{A}} = PA, i.e., collection plus an initial amount of induction together with classical logic yields full induction. Within intuitionistic logic this is different. By means of a functional interpretation of iΔ_{0} + {Bψ | ψ ∈ L_{A}} in a weak fragment of Gödel’s T we establish:

Theorem 7 Suppose φ(x, y) is any formula of L_{A} and iΔ_{0} + {Bψ | ψ ∈ L_{A}} ⊢ ∀x∃yφ(x, y). Then a polynomial p(x) exists such that ∀x∃y ⩽ p(x) φ(x, y).

Finally we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence

Theorem 8 iΔ_{0} + (Coll) ⊢ PA.

Preliminaries

2.a Predicate logic

Formulae of first-order predicate-logic with equality are built up from atomic formulae (this includes equations and ⊥) and the logical connectives ∧, ∨, →, ∀, ∃. We use lower Greek letters φ, ψ, χ … to denote formulae. ¬φ is an abbreviation for φ → ⊥. We fix a Hilbertstyle calculus for intuitionistic first-order predicate logic with equality (see [Tr73], p. 3-4). We have propositional axioms and rules A1 φ → φ A2 φ, φ → ψ ⊢ ψ A3 φ → ψ, ψ → χ ⊢ φ → χ A4 φ ∧ ψ → φ A5 φ ∧ ψ → χ ⊢ φ → ψ → χ A6 φ → ψ → χ ⊢ φ ∧ ψ → χ A7 φ → ψ, φ → χ ⊢ φ → ψ ∧ χ A8 φ → φ ∨ ψ A9 ψ → φ ∨ ψ A10 φ → χ, ψ → χ ⊢ φ ∨ ψ → χ, A11 ⊥ → φ equality axioms

a = a; a = b → φ(a) → φ(b)

and finally Q1 ∀uφ(u) → φ(a); Q2 φ(a) → ∃uφ(u); a a term substitutable in φ

Q3 φ → ψ(u) ⊢ φ → ∀uψ(u); Q4 φ(u) → ψ ⊢ ∃uφ(u) → ψ; u not free in φ (resp. ψ). In Q3 and Q4 the variable u is called eigenvariable. We recall the well-known syntactical abbreviation

∃!yφ(y) :≡ ∃yφ(y) ∧ ∀y_{1}y_{2}(φ(y_{1}) ∧ φ(y_{2}) → y_{1} = y_{2}).

When is comes to classical predicate logic we may restrict to the {∨, ¬, ∀}-fragment of predicate logic and consider →, ∧ and ∃ as defined according to the usual rules φ ∧ ψ :≡ ¬(¬φ ∨ ¬ψ); φ → ψ :≡ ¬φ ∨ ψ; ∃xφ :≡ ¬∀x¬φ. This is a complete calculus for classical predicate logic (see [Sh67] p. 21)

(i) Axioms :

Logical axioms ¬φ ∨ φ
(Substitution) ∀uφ(u) → φ(a) a substitutable in φ
(Equality)  a = a, a = b → φ(a) → φ(b) for atomic φ

(ii) Rules :

(Expansion)  φ ⊢ φ ∨ ψ
(Contraction) φ ∨ φ ⊢ φ
(Associativity) φ ∨ (ψ ∨ χ) ⊢ (φ ∨ ψ) ∨ χ
(∀-Introduction) φ(u) ∨ ψ ⊢ ∀uφ(u) ∨ ψ  (a ∉ ψ)
(Cut)  φ ∨ ψ, ¬φ ∨ χ ⊢ ψ ∨ χ

The eigenvariable of ∀-introduction is u.

2.b Set theory

Let L_{∈} denote the first order language of set-theory with identity and the only non-logical binary predicate symbol ∈. As usual we call quantifiers of the shape ∀x(x^{∈}a → φ) and ∃x(x^{∈}a ∧ φ) restricted quantifiers and abbreviate them by (∀x^{∈}a)φ and (∃x^{∈}a)φ resp.

Definition 2.1 (The formula class Δ_{0}) (i) ⊥, a = b, a^{∈}b are Δ_{0}-formulae

(ii) If φ, ψ are in Δ_{0}, then also (φ ∧ ψ), (φ → ψ), (φ ∨ ψ)

(iii) If φ is in Δ_{0}, then also (∀x^{∈}a)φ and (∃x^{∈}a).

2.b.1 Aczel’s CZF

CZF is formulated in L_{∈}. Its logic is intuitionistic predicate logic and the non-logical axioms and rules of CZF are:

(Extensionality) :  (∀x^{∈}a)(x^{∈}b) ∧ (∀x^{∈}b)(x^{∈}a) → a = b
(Pair) :  ∃z (a^{∈}z ∧ b^{∈}z)
(Union) :  ∃z (∀y^{∈}a)(∀x^{∈}y)(x^{∈}z)
(Infinity) :  ∃x∀y[y^{∈}x ↔ (∀z^{∈}y)(⊥) ∨ (∃u^{∈}x)(suc(u, y))]
(Foundation) :  (∀u^{∈}a)φ(u) → φ(a) ⊢ φ(b)  a ∉ F V (φ)
(Δ_{0}-Separation) :  ∃z∀y(y^{∈}z ↔ y^{∈}a ∧ φ(y))  φ ∈ Δ_{0}
(Strong-Collection) : (∀x^{∈}a)∃wφ(x, w) → ∃zφ′(a, z)
(Subset-Collection) : ∃d ∀u[(∀x^{∈}a)(∃y^{∈}b)φ(x, y) → (∃c^{∈}d)φ′(a, c)].

where suc(x, y) :≡ (∀z^{∈}y)(z = x ∨ z^{∈}x) ∧ x ⊆ y ∧ x^{∈}y and φ′(a, z) :≡ (∀x^{∈}a)(∃y^{∈}z)φ(x, y) ∧ (∀y^{∈}z)(∃x^{∈}a)φ(x, y). Note that in (Subset-Collection) the formula φ may contain u. In (Foundation) the variable a is the eigenvariable.

CZF^{−}denotes CZF without (Subset-Collection). The restricted excluded middle is

(REM) φ ∨ ¬φ φ ∈ Δ_{0}.

We employ the following standard Δ_{0}-definable L_{∈}-predicates:

a ⊆ b, a = {b}, a = (b, c) (the Kuratowski-pair,) Rel(R), fun(f), a = dom(f),
a = rng(f), f(a) = b, a = b ∪ c, a = b ∩ c, a = ⋃ c, a = b − c

and further denote by a × b the Cartesian product of two sets and by ^{a}b the class of all functions f : a → b.

Lemma 2.2 (see [Ac78], propositions 2.1 and 2.2.) We have

CZF^{−} ⊢ ∀ab∃c Full(c, a, b) ↔ (Subset-Collection)
CZF ⊢ ∀ab∃c(c = ^{a}b).

where

Full(c, a, b) :≡ ∀R(R^{∈} mv(^{a}b) → (∃d^{∈}c)R : a >−< d)

R^{∈} mv(^{a}b) :≡ R ⊆ (a × b) ∧ (∀x^{∈}a)(∃y^{∈}b)(x, y)^{∈}R

R : a >−< b :≡ (∀x^{∈}a)(∃y^{∈}b)(x, y)^{∈}R ∧ (∀y^{∈}b)(∃x^{∈}a)(x, y)^{∈}R.

More information about CZF can be found for example in [Ac78].

2.b.2 Kripke-Platek set theory

The following theories are L_{∈}-theories equipped with classical predicate logic. Kripke-Platek set theory with infinity (KPω) contains the following non-logic axioms and rules: (Extensionality), (Pair), (Union), (Infinity), (Δ_{0}-Separation), (Foundation) and

(Δ_{0}-Collection) (∀x^{∈}a)∃yφ(x, y) → ∃w(∀x^{∈}a)(∃y^{∈}w)φ(x, y),

where φ is a Δ_{0}-formula. The standard source for KPω is [Ba75]. To introduce the theories KPi (i for inaccessible) and KPM (M for Mahlo) we add a new unary predicate symbol Ad to the language L_{∈}. Ad is considered to be a Δ_{0}-formula, hence the notion of Δ_{0} changes. The axioms for Ad guarantee that Ad axiomatizes admissible sets:

(Ad1) Ad(u) → ω^{∈}u ∧ tran(u)

(Ad2) Ad(u) ∧ Ad(v) → u^{∈}v ∨ u = v ∨ v^{∈}u

(Ad3) Ad(u) → (Pair)^{u} ∧ (Union)^{u} ∧ (Δ_{0}-Separation)^{u} ∧ (Δ_{0}-Collection)^{u},

where tran(u) :≡ (∀y^{∈}u)(∀z^{∈}y)(z^{∈}u) and φ^{u} is the formula that is obtained by replacing all unrestricted quantifiers ∀x, ∃x in φ by (∀x^{∈}u) and (∃x^{∈}u) resp. ω^{∈}u abbreviates that u contains an infinite set. The theory KPi contains all axioms of KPω, Ad 1 − Ad 3 and further the axiom

(Lim) ∀x∃y(Ad(y) ∧ x^{∈}y).

KPM, in addition, contains the following Mahlo-rule for φ ∈ Δ_{0}:

(Mahlo) ψ ∨ ∃yφ(a, y) ⊢ ψ ∨ ∃z[Ad(z) ∧ (∀x^{∈}z)(∃y^{∈}z)φ(x, y)].

For more information on KPi and KPM see for example [Buc90] and [Ra91].

2.c Arithmetic

The first-order language L_{A} of arithmetic contains non-logical symbols 0, suc, +, ·, <. We put

(∀x<t) ψ :≡ ∀x(x < t → ψ) and (∃x<t) ψ :≡ ∃x(x < t ∧ ψ) x ∉ F V (t)

and call these quantifiers restricted quantifiers. We further write

(∀x ⩽ t) φ :≡ (∀x < suc t) φ, (∃x ⩽ t)φ :≡ (∃x < suc t)φ.

Δ_{0} denotes the class of formulae not containing unrestricted quantifiers. Σ_{0} := Π_{0} := Δ_{0}; the formula classes Π_{n}(Σ_{n}) are defined as usual: n alternating quantifiers starting with an universal (existential) quantifier followed by a Δ_{0}-matrix. The induction axiom Iφ for a formula φ is

Iφ :≡ φ(0) ∧ ∀x(φ(x) → φ(suc x)) → ∀xφ(x).

iΔ_{0} is the intuitionistic L_{A}-theory with defining axioms for 0, suc, +, ·, < (i.e. the axioms of PA^{−}, cf. [Ka91], page 16-18 with obvious bounds for the quantifiers), intuitionistic first order predicate logic (A1-A11,Q1-Q4) with identity and the induction schema Iφ for φ ∈ Δ_{0}. For φ, ψ with x ∉ F V (φ, ψ) we have:

iΔ_{0} ⊢ φ ∨ ψ ↔ (∃x ⩽ 1)[(x = 0 → φ) ∧ (x = 1 → ψ)]

and further

iΔ_{0} ⊢ 0 = 1 ↔ ⊥.

In all theories that contain iΔ_{0} disjunction and ⊥ are definable. Δ_{0}-formulae are decidable in iΔ_{0}, i.e., for φ ∈ Δ_{0} we have

iΔ_{0} ⊢ φ ∨ ¬φ.

A formula φ is called almost negative if it is ∨-free and all its existential quantifiers occur only in front of Δ_{0}-formulae. Suppose Γ is a class of L_{A}-formulae. Then iΓ denote iΔ_{0} plus induction-schema for all φ ∈ Γ. HA (Heyting arithmetic) denotes iΔ_{0} with induction for arbitrary formulae. IΔ_{0}, IΓ and PA (Peano arithmetic) denote the classical counterparts of iΔ_{0}, iΓ and HA resp.

PA and HA prove the same Π_{2}-sentences, the same holds for iΔ_{0} and IΔ_{0}. We sketch the proof of these results:

Definition 2.3 (Negative translation) The negative translation assigns to any formula φ ∈ L_{A} an ∃- and ∨-free formula φ^{−}: (i) s = t^{−} :≡ s = t. (ii) (φ_{0} ∧ φ_{1})^{−} :≡ φ_{0}^{−} ∧ φ_{1}^{−}; (φ_{0} → φ_{1})^{−} :≡ φ_{0}^{−} → φ_{1}^{−}; (φ_{0} ∨ φ_{1})^{−} :≡ ¬(¬φ_{0}^{−} ∧ ¬φ_{1}^{−}) (iii) (∀xφ_{0})^{−} :≡ ∀xφ_{0}^{−}; (∃xφ_{0})^{−} :≡ ¬∀x¬φ_{0}^{−}

Definition 2.4 (Friedman (Dragalin) translation) For formulae φ and ψ (with the proviso that no free variable of ψ is bound in φ) we define by induction on φ the Friedman-translation φ^{ψ}:

(i) (s = t)^{ψ} :≡ s = t ∨ ψ (ii) (·)^{ψ} commutes with all logical connectives.

Lemma 2.5 For φ ∈ Δ_{0} we have (i) iΔ_{0} ⊢ φ^{−} ↔ φ (ii) iΔ_{0} ⊢ (φ)^{ψ} ↔ (φ ∨ ψ) □

Theorem 2.6 (see [TvD88 ], vol. I, p. 58 and p. 137) Suppose Γ ⊆ L_{A}. Define Γ^{−} := {φ^{−} | φ ∈ Γ} and Γ^{ψ} := {φ^{ψ} | φ ∈ Γ}. Then we have IΓ ⊢ φ ⇒ i(Γ^{−})^{ψ} ⊢ (φ^{−})^{ψ}. □

Corollary 2.7 (i) PA is Π_{2}-conservative over HA. (ii) IΔ_{0} is Π_{2}-conservative over iΔ_{0}. □

Remark 2.8 Note that this proof essentially depends on properties of the formula classes L_{A} and Δ_{0}. For other classes like Σ_{1} this method does not apply.

2.d Functional interpretation of arithmetic

2.d.1 The Dialectica interpretation of HA

Definition 2.9 (Finite types) The collection T of finite linear type symbols and their type rank are defined by:

(i) o ∈ T ; rk o = 0 (ii) σ, τ ∈ T ⇒ (σ → τ) ∈ T ; rk(σ → τ) := max{rk(σ) + 1, rk(τ)}. We usually suppress the outermost brackets and abbreviate 1 :≡ o → o, 2 :≡ 1 → o, σ → τ → ρ :≡ σ → (τ → ρ).

Definition 2.10 (Type structure) A type structure M = (M_{σ} | σ ∈ T ) is a family of non-empty sets M_{σ} for σ ∈ T where (i) M_{o} is any nonempty set (ii) M_{σ→τ} is a set of functions (or codes of functions) from M_{σ} to M_{τ}. In the context of arithmetic the ground type o denotes the set of natural numbers, the types of the form σ → τ are intended to range over functions from ℕ_{σ} to ℕ_{τ} and the standard structure is the full type structure over the natural numbers:

Definition 2.11 (The full type structure over ℕ) N = (ℕ_{σ} | σ^{∈}T ), where (i) ℕ_{0} := ℕ (ii) ℕ_{σ→τ} := {f | f : ℕ_{σ} → ℕ_{τ}}. This is Gödel’s system T of primitive recursive functionals:

Definition 2.12 (Gödel’s T) The language L(T) contains the following basic functionals (we write a : τ to indicate that a is of type τ):

- 0 : o

- suc : 1

- K_{στ} : σ → τ → σ for arbitrary σ, τ

- S_{ρστ} : (ρ → σ → τ) → (ρ → σ) → ρ → τ for arbitrary ρ, σ, τ

- R_{τ} : τ → (o → τ → τ) → o → τ. L(T) contains countably many variables x^{τ} : τ for any type τ. Terms of L(T) and their types are defined inductively:

(i) Basic functionals and variables are terms of the given type.

(ii) If a : σ → τ and b : σ then (ab) : τ.

Formulae of L(T) are defined inductively:

(i) ⊥ is an L(T)-formula; a = b is an L(T)-formula if a, b are terms of the same type.

(ii) If φ, ψ are L(T)-formulae, then (φ → ψ), (φ ∧ ψ) and (φ ∨ ψ) are L(T)-formulae.

We define rk a := rk τ for any term a : τ. Formulae of type o are those built up from equations a^{o} = b^{o} using → and ∧. T contains positive sentential logical axioms and rules A1 − A11 and the following defining equations for the basic functionals:

(i) Kab = a,

(ii) Sabc = ac(bc)

(iii) Rab0 = a, Rab(suc s) = bs(Rabs).

We further have an identity schema

a = b → φ(a) → φ(b)

and the induction rule for all L(T)-formula φ with x ∉ F V (φ) and t substitutable in φ:

φ[0], φ[x] → φ[suc x] ⊢ φ[t].

Convention 2.13 We use letters a, b, s, t, … to denote sequences of variables or terms and introduce the following abbreviations:

\begin{aligned}ab & :≡ ab_{1} … b_{n} & for a ∈ τ_{1} → … → τ_{m} → σ, b_{i} ∈ τ_{i} (i = 1, …, n) \\ ab & :≡ a_{1}b, …, a_{n}b & for a_{i} ∈ τ_{1} → … → τ_{m} → σ_{i}, b_{i} ∈ τ_{i} (i = 1, …, n) \end{aligned}

An essential property of T is that we have characteristic terms for formulae of type o:

Lemma 2.14 (Characteristic terms) For all formulae φ of T of type o we have a term χ_{φ} in T, with the same free variables as φ such that T ⊢ φ ↔ χ_{φ} = 1. □

Definition 2.15 (The Dialectica translation) The Dialectica translation assigns inductively a formula ∃v∀wφ_{D} with φ_{D} ∈ L(T) to every formula φ ∈ L_{A} (v, w of arbitrary finite type and F V (φ) = F V (∃v∀wφ_{D})):

2.d Functional interpretation of arithmetic  15

(i) s = t^{D} :≡ s = t, ⊥^{D} :≡ ⊥ Suppose φ^{D} ≡ ∃v∀wφ_{D}, ψ^{D} ≡ ∃y∀zψ_{D} with disjoint sequences of variables vwyz.

(ii) (φ ∧ ψ)^{D} :≡ ∃vy∀wz(φ_{D} ∧ ψ_{D}) (iii) (φ ∨ ψ)^{D} :≡ ∃vyt∀wz[(t = 0 → φ_{D}) ∧ (t ≠ 0 → ψ_{D})] (iv) (φ → ψ)^{D} :≡ ∃W Y ∀vz[φ_{D}(v, W vz) → ψ_{D}(Y v, z)]

(v) (∃uφ)^{D} :≡ ∃uv∀wφ_{D} (vi) (∀uφ)^{D} :≡ ∃V ∀uwφ_{D}(V u, w, u). A formula φ ∈ L_{A} with φ^{D} ≡ ∃v∀wφ_{D} is said to be^{D}-interpretable in T if there is a sequence of terms t ∈ T containing at most the free variables of φ such that T ⊢ φ_{D}(t, w). Note that the matrix φ_{D} is a formula of type o whenever φ ∈ L_{A}: hence there is a characteristic term in T for each φ_{D} provided φ ∈ L_{A}.

Theorem 2.16 (Gödel’s Dialectica interpretation, [G58]) HA is ^{D}-interpretable in T. □

2.d.2 The Shoenfield interpretation of PA

The Shoenfield interpretation is a modification of the Dialectica interpretation that is directly applicable to PA. Since PA is provided with classical we can tend to the {¬, ∨, ∀}-fragment of L_{A}.

Definition 2.17 (The Shoenfield translation) The Shoenfield translation ^{S} assigns inductively a formula ∀v∃wφ_{S} to every φ of PA. Here v and w are tuples of variables of finite type.

(i) (a = b)^{S} :≡ a = b Suppose now φ^{S} ≡ ∀v∃wφ_{S} and ψ^{S} ≡ ∀y∃zψ_{S} with sequences of distinct variables.

(ii) (φ ∨ ψ)^{S} ≡ ∀vy∃wz(φ_{S} ∨ ψ_{S}) (iii) (∀uφ)^{S} ≡ ∀uv∃wφ_{S} (iv) (¬φ)^{S} ≡ ∀W ∃v¬φ_{S}(v, W v)

A formula φ with ^{S}-translation ∀v∃wφ_{S} is ^{S}-interpretable in T if there are terms t in T containing at most the free variables of φ and v such that T ⊢ φ_{S}(v, t).

Shoenfield’s interpretation (cf. [Sh67]) is performed in the full type structure, but of course the following holds as well:

Theorem 2.18 (The Shoenfield interpretation, [Sh67 ] p. 220) PA is ^{S}-interpretable in T. □

2.e Arithmetic in all finite types

The natural framework to discuss the strength of the Dialectica translation, i.e., to investigate the schema φ ↔ φ^{D}, is Heyting arithmetic in all finite types.

Definition 2.19 (HA^{ω}, HA^{ω+}) The language L_{A}^{ω} of HA^{ω} is L(T) extended by quantifiers for all finite types. Types, terms and prime formulae of HA^{ω} are those of T. The class of formulae is defined by the following clauses

(i) Prime formulae are formulae

(ii) If φ and ψ are formulae then (φ∧ψ), (φ∨ψ), (φ → ψ), (∀x^{σ}φ) and (∃x^{σ}φ) are formulae.

Axioms and rules of HA^{ω} are those of T plus Q1 − Q4. We consider the following additional axiom schemes:

(AC_{στ}) ∀x^{σ}∃y^{τ}φ(x, y) → ∃Y^{σ→τ}∀x^{σ}φ(x, Y x); (AC) := ⋃_{σ,τ} AC_{στ}
(M)  ∀w(φ ∨ ¬φ) → ¬∀wφ → ∃w¬φ
(IP)  ∀w(φ ∨ ¬φ) → (∀wφ → ∃yψ) → ∃y(∀wφ → ψ)

and define HA^{ω∗} := HA^{ω} + (AC) + (IP) + (M).

Remark 2.20 HA is not literally a fragment of HA^{ω}. Of course +, ·, < are definable in HA^{ω} and hence we may consider HA as an subsystem of HA^{ω}.

Definition 2.21 (PA^{ω}) PA^{ω} is HA^{ω} with classical predicate logic.

Theorem 2.22 (Characterization of ^{D}, see [Tr73] p. 239) HA^{ω∗} = HA^{ω} + {φ ↔ φ^{D} | φ ∈ L_{A}^{ω}}. □

Theorem 2.23 (Characterization of ^{S}) PA^{ω} + {∀x^{σ}∃y^{τ}φ(x, y) → ∃Y^{σ→τ}∀x^{σ}φ(x, Y x) | φ ∈ Δ_{0}} = PA^{ω} + {φ ↔ φ^{S} | φ ∈ L_{A}^{ω}}. □

It is clear that due to the presence of equations of arbitrary type we do not have characteristic terms even for atomic formulae of HA^{ω}. So the interpretation of A7 with characteristic terms fails. Even more: HA^{ω} is not ^{D}-interpretable in T:

Example 2.24 (Howard, see [DN74 ]) Let 0^{oo} :≡ K_{o→o}0 and consider

∀u^{o→o}¬¬∃x(x = 0 ↔ u^{o→o} = 0^{o→o}).

This formula is derivable in HA^{ω} but not ^{D}-interpretable in T. Otherwise there would be a type 2 functional Y ∈ T such that

(1) ¬¬(Y u = 0 ↔ u = 0^{o→o}).

Hence we have Y 0^{o→o} = 0. All functionals of T of type 2 are continuous, i.e., we have

(2) ∀v^{o→o}∃z^{o}∀u^{o→o}((∀x<z)(ux = vx) → Y u = Y v).

This implies

∃z∀u((∀x<z)ux = 0 → Y u = 0)

and this contradicts (1).

2.e.1 The Diller-Nahm interpretation of HA^{ω}

Diller and Nahm in [DN74] introduced the system T_{∧} and the translation ^{∧} to overcome the contraction problem. T_{∧} extends Gödel’s T in the following way:

Definition 2.25 (T_{∧}) The language of T_{∧} arises form L(T) by adding a restricted universal quantifier, i.e., in the inductive definition of the formulae of T_{∧} we add the following clause:

(i) if φ is a formula of T_{∧} and t a term of type o that does not contain the free variable x : o then (∀x<t)φ is a formula of T_{∧}. Formally (∀x<t)φ abbreviates ∀x(x < t → φ).

The restricted universal quantifier is axiomatized by the following four axioms:

B1 (∀x<0)φ B2 (∀x<suc t)φ(x) → (∀x<t)φ(x) B3 (∀x<suc t)φ(x) → φ(t) B4 (∀x<t)φ(x) ∧ φ(t) → (∀x<suc t)φ(x). The Diller-Nahm translation differs from the Dialectica translation only in the case of the implication. Note that we now treat the restricted universal quantifier as a separate logical operator.

Definition 2.26 (The Diller-Nahm translation, [DN74 ] p. 54-55) The ^{∧}-translation assigns inductively a formula ∃v∀wφ_{∧} with φ_{∧} ∈ L(T_{∧}) to every formula φ in the language of HA^{ω} (v, w of finite type and F V (φ) = F V (∃v∀wφ_{∧})):

(i) s = t^{∧} :≡ s = t ⊥^{∧} :≡ ⊥ Suppose φ^{∧} ≡ ∃v∀wφ_{∧}, ψ^{∧} ≡ ∃y∀zψ_{∧} with disjoint sequences of variables vwyz.

(ii) (φ ∧ ψ)^{∧} :≡ ∃vy∀wz[φ_{∧} ∧ ψ_{∧}] (iii) (φ ∨ ψ)^{∧} :≡ ∃vyt∀wz[(t = 0 → φ_{∧}) ∧ (t ≠ 0 → ψ_{∧})] (iv) ((∀x<t)φ)^{∧} :≡ ∃V ∀z[(∀x<t)φ_{∧}(Y x, z)]

(v) (φ → ψ)^{∧} :≡ ∃W XY ∀vz[(∀x<Xvz)φ_{∧}(v, W vzx) → ψ_{∧}(Y v, z)] if w is not the empty tuple.

(vi) (φ → ψ)^{∧} :≡ ∃Y ∀vz(φ_{∧} → ψ_{∧}(Y v, z)) if w is the empty tuple (vii) (∃uφ)^{∧} :≡ ∃uv∀wφ_{∧} (viii) (∀uφ)^{∧} :≡ ∃V ∀uwφ_{∧}(V u, w, u), if ∀u is not a restricted quantifier.

Theorem 2.27 (Diller-Nahm interpretation, [DN74 ] p. 57) HA^{ω} + {φ ↔ φ^{∧} | φ ∈ L_{A}^{ω}} is ^{∧}-interpretable in T_{∧}. □ It remains to introduce the principles that are needed to characterize the translation ^{∧}, these are (AC) again and modified versions of (M) and (IP):

(IP_{∧}) :≡ (∀wφ → ∃yψ) → ∃y(∀wφ → ψ), where y is a tuple of variables not free in φ and φ ∈ T_{∧}.

(M_{∧}) :≡ (∀wφ → ψ) → ∃W X[(∀x<X)φ(W x) → ψ], where the variables x, X and W are not free in φ, ψ and φ, ψ ∈ T_{∧}.

HA^{ω+} := HA^{ω} + (AC) + (IP_{∧}) + (M_{∧}).

Theorem 2.28 (Characterization of ^{∧}, [DN74 ] p. 56) HA^{ω+} = HA^{ω} + {φ ↔ φ^{∧} | φ ∈ L_{A}^{ω}}. □

2.e.2 Fragments HA_{0}^{ω}, T_{0}

Definition 2.29 (HA_{0}^{ω}, T_{0}) T_{0} is the fragment of T by taking out equations of type τ ≠ o. Since the defining axioms for K, S and R (they are in general equations of finite type) are no longer in this fragment, we have to use the corresponding replacement schemas to axiomatize K, S and R. For arbitrary terms t[x] of type o we have axioms

(i) t[Kab] = t[a]

(ii) t[Sabc] = t[ac(bc)]

(iii) t[Rab0] = t[a]; t[Rab(suc s)] = t[bs(Rabs)].

In a similar way we define the fragment HA_{0}^{ω}: again we have only equations of type o.

Quantifier free formulae of HA_{0}^{ω} are decidable and we have characteristic terms for these formulae in T_{0}. Hence we have

Theorem 2.30 HA_{0}^{ω} is ^{D}-interpretable in T_{0}. □

2.e.3 Hereditarily majorizable functionals

In [Ho73] Howard shows that the simplest non-trivial case of the axiom of extensionality, namely

(E2) ∀u(αu = βu) → Y α = Y β,

(where Y : 2, α, β : 1) is not ^{D}-interpretable in T (it is as well not ^{∧}-interpretable in T_{∧}). For this purpose he introduces the notion of hereditarily majorizability:

Definition 2.31 (Hereditarily majorizable functionals) By induction on σ we define X^{∗} maj_{σ} X for functionals X^{∗}, X of type σ:

(i) a^{∗} maj_{o} a :≡ a^{∗} ⩾ a

(ii) F^{∗} maj_{σ→τ} F :≡ ∀GG^{∗}(G^{∗} maj_{σ} G ⇒ F^{∗}G^{∗} maj_{τ} F G)

A functional F : τ is called hereditarily majorizable if there is a functional F^{∗} : τ such that F^{∗} maj_{τ} F.

Theorem 2.32 ([Ho73 ]) (i) All functionals of T are hereditarily majorizable.

(ii) No hereditarily majorizable functional ^{D}-interprets (E2) in T.

(iii) Hence (E2) is not ^{D}-interpretable in T. □

2.e.4 Equality in higher types

Equality in higher types is a delicate point and there are several different ways to axiomatize it. Unfortunately there is no standard notation in the literature concerning this point. We tend to [Tr90] and describe the following variants of HA^{ω} (in [Tr73] this system is called N − HA^{ω}, where N stands for neutral):

Definition 2.33 (I − HA^{ω}) The extension of HA^{ω} by adding characteristic functionals E_{σ} for equality of type σ for each σ and axioms

E_{σ}(f^{σ}, g^{σ}) = 0 ↔ f = g.

is called intensional I − HA^{ω}

Theorem 2.34 I − HA^{ω} is ^{D}-interpretable in T + {E_{σ}(f^{σ}, g^{σ}) = 0 ↔ f = g | σ ∈ T }. □

Note that T augmented with equality functionals is a quite different system than T. While all type 2 functionals of T are continuous, this does not longer hold for the extended system.

Definition 2.35 (E − HA^{ω}) The extensional variant E − HA^{ω} is formulated in the language of HA_{0}^{ω} with defined extensional equality for all types σ ≠ o:

s = t :≡ ∀x_{1} … x_{n}(sx_{1} … x_{n} = tx_{1} … x_{n})

where sx_{1} … x_{n} is of type o. E − HA is equipped with the equality schema for this defined equality as well.

E − HA^{ω} ⊢ (E2) and hence 2.32 gives

Theorem 2.36 E − HA^{ω} is not ^{D}-interpretable in T (it is also not ^{∧}-interpretable in T_{∧}). □

An intermediate system is the following:

Definition 2.37 (WE − HA^{ω}) The theory WE − HA ^{ω} is similar to E − HA^{ω} but instead of the full equality schema for the defined equality we have only a weaker rule

(Ext-Rule) tx_{1} … x_{n} = sx_{1} … x_{n}, φ(t) ⊢ φ(s).

Due to the fact that the Dialectica translation of the (Ext-Rule) is an instance of that rule again we have

Theorem 2.38 WE − HA^{ω} is ^{D}-interpretable in T_{∈} + (Ext-Rule). □

Lemma 2.39 ([Tr73], p. 242) The deduction theorem does not hold for WE − HA^{ω}. □

Part I Set Theory Towards a functional interpretation of set theory

3.a Concepts and aims of functional interpretations

In the first part of this thesis we give functional interpretations of constructive set theory and Kripke-Platek set theory by means of refined and modified versions of Gödel’s Dialectica interpretation of Heyting arithmetic (HA). We start with some remarks on Gödel’s Dialectica interpretation. However, this section is not intended to be a complete survey of this subject (here we recommend Avigad and Feferman [AF98], Troelstra [Tr90] and Troelstra [Tr73] for example). It focuses rather on aspects related to problems that arise in the context of set theory.

When we discuss the origins of Gödel’s Dialectica interpretation, we start with Hilbert’s program. As Hilbert never formulated it exactly, we may roughly summarize it as follows:

- Formalize mathematics in a formal system.

- Show the consistency of this system with finitist methods.

Hilbert formulated this program in the twenties and by finitist methods he meant mathematics that deals only with concrete finite objects and combinatorial operations on them. Primitive recursive arithmetic for example is a suitable system to carry through Hilbert’s program. Gödel’s second incompleteness theorem (1931) showed that this program was unattainable: No consistent, recursively axiomatized system that contains primitive recursive arithmetic proves its own consistency, let alone the consistency of Peano arithmetic or even stronger theories. Hence any consistency proof of classical mathematics or even arithmetic exceeds the framework of finitary mathematics. Gödel’s famous paper from 1958 ’Über eine bisher noch nicht benütze Erweiterung des finiten Standpunktes’ [G58] starts with the following comment on this fact:

P. Bernays hat wiederholt darauf hingewiesen, dass angesichts der Tatsache der Unbeweisbarkeit der Widerspruchsfreiheit eines Systems mit geringeren Beweismitteln als denen des Systems selbst eine Überschreitung des Rahmens der im Hilbertschen Sinn finiten Mathematik nötig ist, um die Widerspruchsfreiheit der klassischen Mathematik, ja sogar um die der klassischen Zahlentheorie zu beweisen. Da die finite Mathematik als die der anschaulichen Evidenz definiert ist, so bedeutet das [ …], dass man für den Widerspruchsfreiheitsbeweis der Zahlentheorie gewisse abstrakte Begriffe braucht.^{1}

Gödel concludes:

Jedenfalls lehrt uns die Bernayssche Bemerkung, zwei Bestandteile in der finiten Einstellung unterscheiden, nämlich erstens das konstruktive Element, welches darin besteht, dass von mathematischen Objekten nur insoweit die Rede sein darf, als man sie aufweisen oder durch Konstruktion tatsächlich herstellen kann; zweitens das spezifisch finitistische Element, welches darüber hinaus fordert, dass die Objekte, über welche man Aussagen macht, mit welchen die Konstruktionen ausgeführt werden und welche man durch sie erhält, “anschaulich” sind, das heisst letzten Endes raum-zeitliche Anordnungen von Elementen, deren Beschaffenheit abgesehen von Gleichheit und Verschiedenheit irrelevant ist.^{2}

In view of his second incompleteness theorem, Gödel continues to argue that in order to realize at least a relativised Hilbert program one has to drop the second requirement: Certain abstract notions have to be accepted. A prominent example is Gentzen’s consistency proof of Peano arithmetic (PA). Here ordinals up to ε_{0} are adjoint to finitary mathematics. In his paper from 1958 Gödel takes another route and shows that the notion of computable or primitive recursive functional of finite type is sufficient for that task as well. (Functionals of finite type were introduced first by Hilbert in 1925, see [Hi26].) For that reason Gödel introduces a quantifier free term calculus T of primitive recursive functionals of finite type which generalize the primitive recursive functions. In particular, T allows the definition of a functional F from given functionals G and H by primitive recursion:

1English translation by Stefan Bauer-Mengelberg and Jean van Heijenoort in [G90] p. 241: P. Bernays has pointed out on several occasions that, since the consistency of a system cannot be proved using means of proof weaker than those of the system itself, it is necessary to go beyond the framework of what is, in Hilbert’s sense, finitary mathematics if one wants to prove consistency of classical mathematics, or even that of classical number theory. Consequently, since finitary mathematics is defined as the mathematics in which evidence rests on what is intuitive, certain abstract notions are required for the proof of the consistency of number theory [ …].

2At the same place, p. 245: In any case Bernays’ remark teaches us to distinguish two components in the finitary attitude; namely, first, the constructive element, which consists in our being allowed to speak of mathematical objects only in so far as we can exhibit them or actually produce them by means of a construction; second, the specifically finitistic element, which makes the further demand that the objects about which we make statements, with which the constructions are carried out and which we obtain by means of these constructions, are ‘intuitive’, that is, are in the last analysis spatiotemporal arrangements of elements whose characteristics other than their identity or nonidentity are irrelevant.

F (0) = G F (x + 1) = H(x, G(x)).

In contrast to the class of primitive recursive functions, G and H may be of finite type – this allows us to define much more complex functions, e.g. the Ackermann function and, in fact, all ≺ ε_{0}-descent recursive functions. T further contains the constant 0, a successor functional suc and is combinatorially complete, i.e., allows explicit definition. Furthermore, it is equipped with an induction rule. The main result of Gödel’s paper is an interpretation of HA in T: Gödel assigns to any formula φ in the language of HA a formula φ^{D} of the form ∃v∀wφ_{D}(v, w) where φ_{D} is a formula of T and v, w are tuples of variables of finite type – this translation is called the Dialectica translation. Gödel proves: Whenever HA ⊢ φ with translation φ^{D} ≡ ∃v∀wφ_{D}(v, w) then there are terms t in T such that T ⊢ φ_{D}(t, w), or, in other words: φ is ^{D}-interpretable in T.

In the translation ^{D} the case of implication is of particular interest. In his paper from 1958, Gödel gives the definition of ^{D} without any explanation. However, in earlier sources we find a detailed description of how to perform and justify this transition. Suppose we are given two expressions ∃x∀yM(x, y) and ∃u∀vN(u, v) with φ, ψ quantifier free. The task is to find an expression which is again in ∃∀-shape and has ‘the same meaning’ as the implication

(1) (∃x)(∀y)M(x, y) → (∃u)(∀v)N(u, v).

These are Gödel’s explanations (quoted from the talk ‘In what sense is intuitionistic logic constructive’ held by Gödel at Yale university in 1941 [G94], page 196-197, we adapt Gödel’s notation and write (∀x) instead of (x))^{3}:

This implication [i.e. (1)] means: If there exists an object x satisfying a certain condition then there exists also an object u satisfying a certain other condition.

3It is remarkable that in his 1941 talk we already find all the ideas of his 1958 paper. In many aspects this talk gives more detailed explanations than his 1958 paper.

But in a constructive logic that can only mean: Given such an x you can construct such a u, i.e.,

(∃f)(∀x)[(∀y)M(x, y) → (∀v)N(f(x), v)].

But this expression still has not the form we want, since implication still is applied to expressions containing quantifiers. But now we have only universal quantifiers here, and what can in a constructive logic be the meaning of the assertion: If ∀xF (x) then ∀yG(y)? The simplest interpretation which suggests itself is this: Given a counterexample for G you can construct a counterexample for F, i.e., the expression in square brackets will mean:

(∃g)(∀v)[∼ N(f(x), v) →∼ M(x, g(v))];

but here the symbol of implication is applied only to expressions without quantifiers. Therefore, we can replace it by the ordinary implication of two-valued logic. Furthermore, we can apply transposition to this implication, interchanging the order of the two terms, and obtain in this manner for the whole expression:

(∃f)(∀x)(∃g)(∀v)[M(x, g(v)) ⊃ N(f(x), v)].

But now, that for every x there exists such a function g means that g is really a function of two variables x and v, i.e., we finally obtain

(∃f, g)(∀x, v)[M(x, g(x, v)) ⊃ N(f(x), v)],

and now we are through, since this expression has the form we want [ …].

At this point we should note that this justification essentially needs the decidability of quantifier free formulae. The translations of the remaining logical operations are defined and justified in a much simpler way and we do not want to describe them here (see 2.15).

Let us briefly summarize what we gain from this interpretation (cf. Troelstra’s introduction to the 1941 lecture, [G94] p. 187):

- First of all, this reduces Heyting arithmetic to the quantifier free term calculus T. Hence HA is consistent relative to T.

However, there are various further applications of this interpretation:

- Together with the negative-translation, this interpretation also reduces classical Peano arithmetic to the same system T.

- The provably recursive functions of HA are characterized in terms of T: whenever HA ⊢ ∀x∃yφ(x, y) with φ a primitive recursive predicate, then there is a term t in T such that T ⊢ φ(x, tx) (the same holds for PA).

- If HA ⊢ ∃xφ(x) for an arbitrary formula φ with φ^{D} ≡ ∃v∀wφ_{D}, then there are terms t, v in T with T ⊢ φ_{D}(t, v, w) (in general, this does not hold for PA).

- A version of Markov’s rule can be justified for HA: Whenever HA ⊢ ¬¬∃xφ(x) with φ primitive recursive, then T ⊢ φ(t) for a term t and hence by a computability proof within HA we get HA ⊢ ∃xφ(x).

- There is a number-theoretic formula A(x) such that ¬∀x(A(x) ∨ ¬A(x)) can be consistently added to intuitionistic arithmetic HA (see [Tr90], p. 187).

The proof of the interpretation theorem proceeds along the derivations of HA and we want to give some hints of the proof. Induction is interpreted by using the schema of primitive recursion in higher types. The interpretation of the cut rule uses the composition of functionals. The interpretation of modus ponens corresponds to application and for φ → φ∨ψ and φ∧ψ → φ one uses canonical constants for all types. All these operations, namely composition, primitive recursion and application, and the canonical constants are available in the system T. The interpretation of the remaining propositional axioms and the quantifier rules and axioms are straightforward, besides the following innocent-looking axiom (in our calculus this axiom is a special case of A7):

φ → φ ∧ φ.

This is a contraction axiom and we want to take a closer look at its interpretation. For simplicity we assume for the moment that φ ≡ ∀wφ_{0}(w) where φ_{0} is quantifier free. Hence φ^{D} ≡ φ and the translation of the contraction axiom reads

(φ → φ ∧ φ)^{D} ≡ ∃W ∀yz[φ_{0}(W yz) → φ_{0}(y) ∧ φ_{0}(z)].

For the interpretation we need a term t in T such that

(2) φ_{0}(tyz) → φ_{0}(y) ∧ φ_{0}(z).

That means t has to decide, for all given y, z, which one of these two is the essential witness for (2), in particular we may define tyz := z if ¬φ_{0}(y) and tyz := y else ^{4}. The crucial point is that this decision has to be made in terms of T. Provided we have a characteristic term t_{φ_{0}} for φ_{0} with T ⊢ t_{φ_{0}} = 1 ↔ φ_{0} this is possible and we can interpret (2).

The example φ ≡ ∀wφ_{0} already covers the problems of the general case. Hence for arbitrary φ ∈ L_{A} with translation ∃v∀wφ_{D} the matrix φ_{D} has to fulfill these two properties in order to make the argument work:

(i) We need a characteristic term in T for φ_{D}.

(ii) φ_{D} has to be decidable, i.e., T ⊢ φ_{D} ∨ ¬φ_{D}.

As long as we are dealing with HA, both requirements are fulfilled, because for φ ∈ L_{A} the matrix φ_{D} of the Dialectica-translation is a Boolean combination of equations of type o. These formulae are decidable and it is easy to define characteristic terms for them in T. However, there are formulae in T for which these requirements fail, namely equations of higher type. Consider for example φ :≡ x^{1} = y^{1} for variables of type 1. We neither have a characteristic term for φ nor T ⊢ φ ∨ ¬φ. We should mention that the decidability of a formula does not entail the existence of a characteristic term for this formula, here the example is T provided with classical logic: although now all formulae are decidable, we still do not have characteristic terms for equations of higher type. And vice versa, in general the existence of a characteristic term does not imply the decidability of a formula: here the example is the system of constructive set functionals we define in section 4.a. For Δ_{0}- formulae this system is provided with characteristic terms, although even prime formulae are undecidable.

It is remarkable that Gödel does not mention these aspects at all in his paper from 1958. Troelstra comments ([Tr90], p. 228):

It is doubtful whether in 1958 Gödel had already realized the need for characteristic functions in connection with the axiom A → A ∧ A. In a letter of 1970 J. Diller explicitly drew Gödel’s attention to the role of characteristic functions. Gödel reacted in a letter to Bernays (14 July 1970) as follows: “I do not understand what it means to say that in my proof of the formula p ⊃ p ∧ p a passage (which is not possible) to the characteristic term of a formula is required. What is required is the decidability of intensional equations between functions.”

4It is also possible to make the definition the other way round. This shows that the terms of the Dialectica interpretation are not uniquely determined.

Gödel probably regarded the existence of (computable) equality functionals as a concomitant of the decidability of equality; from the passage just quoted one cannot tell whether Gödel realized that in general (that is, at higher types) the decidability of equality does not entail, axiomatically, the presence of equality functionals at higher types.

In 1972, Gödel wrote a second version of his paper from 1958, this time in English ([G90], pp. 271-280)^{5}. There he treats the problems concerning contraction in footnote i2. In an earlier version of that footnote, later crossed out, he introduces equality functionals with axioms G(f, g) = 0 ↔ f = g for all types. Later he realized that the interpretation of HA needs only characteristic terms for formulae of type o and the final version of this footnote is ([G90], p. 277):

For the proofs that the Axioms 1 and 4 of H and the deduction Rule 6 of H^{6} hold in the interpretation [ …], the following principle of disjunctive definition is needed: A function f may be defined by stipulating

A ⊃ f(x) = t_{1}, ¬A ⊃ f(x) = t_{2},

where t_{1}, t_{2} are terms and A is a truth-value function of equations between number terms, both containing only previously defined functions and no variables except those of the sequence x.

We may summarize this discussion: the Dialectica interpretation is applicable to HA only due to the conditions that both the matrices of the translated formulae are decidable and we have characteristic terms for these formulae. None of these conditions follows in general from the other one.

As soon as it comes to extensions of HA, these requirements may fail. One interesting extension is the natural span of HA and T: Heyting arithmetic in all finite types. This theory is the framework to discuss the strength of the Dialectica-translation, i.e., to characterize the schema φ ↔ φ^{D}. There are various versions of arithmetic in all finite types: HA^{ω}, HA_{0}^{ω}, I − HA^{ω}, E − HA^{ω} and WE − HA^{ω}, and they all differ in how equality in higher types is treated. For these systems we refer to section 2.e.4 (see also [Tr90], pp 230-231, [AF98] and [Tr73]. One has to be careful because sometimes these sources differ in notation.) Concerning the Dialectica-interpretability we should mention that HA_{0}^{ω} and WE − HA^{ω} are Dialectica-interpretable in T. I − HA^{ω} is equipped with equality functionals E_{σ} for each type and is ^{D}-interpretable in T + E_{σ}. (This system is quite different from T: the type 2 functionals are no longer continuous and E_{1} is not provably recursive in I − HA^{ω}.) On the other hand, HA^{ω} (in [Tr73] this is called N − HA^{ω}, where N stands for neutral) and E − HA^{ω} are not Dialectica-interpretable, but for quite different reasons. Howard showed that no functional of T ^{D}-interprets the axiom (E2) (cf. 2.e.3) of extensionality for type 2. In E − HA^{ω} equality in higher types is extensional and hence (E2) is derivable. This, of course, implies that E − HA^{ω} is not ^{D}-interpretable.

5The paper was planned to be published in the Swiss journal Dialectica. This never happened and it appeared in [G90] for the first time.

6Axiom 1 is φ → φ ∧ φ, axiom 4 φ ∨ φ → φ and rule 6 is φ → ψ, χ → ψ ⊢ φ ∨ χ → ψ

HA^{ω} is an intensional version where we neither have the decidability of equations of higher types nor additional equality functionals. This example by Howard shows that HA^{ω} is not D-interpretable in T: Let 0^{oo} be a canonical zero-functional of type 1 and φ be the formula

(3) ∀u^{o→o}¬¬∃x(x = 0 ↔ u^{o→o} = 0^{o→o}).

This formula is derivable in HA^{ω} but not ^{D}-interpretable by any continuous functional. Since all functionals of T are continuous we see that HA^{ω} is not^{D}-interpretable in T (for the precise argument see example 2.24). The derivation of (3) essentially makes use of the contraction axiom and in contrast to the non-interpretability of (E2) this example has to do with the lack of characteristic terms. This example demonstrates that in order to interpret the contraction of HA^{ω} the argument sketched above fails: In T we do not have characteristic terms for the prime formulae of HA^{ω}.

In 1974, Diller and Nahm gave a refined translation ^{∧} to overcome this contraction problem and make the interpretation independent of characteristic terms and the decidability of quantifier free formulae. The main idea of the Diller-Nahm translation is the following: The Dialectica translation requires a decision between two witnesses, which is in general impossible. The Diller-Nahm translation, in contrast, allows us to collect both witnesses and hence makes the decision superfluous. We now want to describe the idea of this translation in more detail (see also [DV75]). Again the translation sends any φ (now φ a formula of HA^{ω}) to a formula: φ^{∧} ≡ ∃v∀wφ_{∧}. Almost all cases of the translation are just as in the Dialectica translation (for the precise definition see 2.26). Only the interesting case, namely implication, is different. To motivate this translation, let us start with an implication

(5) ∃v∀wφ → ∃y∀zψ,

where φ and ψ are quantifier free. We are looking for an ∃∀-formula with “almost” the same meaning as (5). Since (5) is intuitionistically equivalent to

(6) ∀v(∀wφ → ∃y∀zψ)

we may treat v as free parameters. This situation is visualized (as a natural deduction) by the following picture:

[figure: ∀wφ[v]; ∀wφ[v]; ∃y∀zψ]

where we may have possibly several assumptions ∀wφ[v]. Heyting arithmetic (also in all finite types) satisfies existential definability (ED), that is: from a derivation of a closed formula ∃vφ one can construct a witness v. This property still holds if the derivation depends on a negative (i.e., ∃- and ∨-free) assumption, which is the case in our situation. This consideration justifies the passage from (6) to ∃y(∀wφ[v] → ∀zψ). Again the step to ∃y∀z(∀wφ[v] → ψ) is valid in intuitionistic logic and after computing y_{0} we may again consider z as free parameters. Hence we have reached the following situation (and until now the procedure is exactly the same as for the Dialectica translation):

(7) ∀wφ[v] → ψ[y_{0}, z],

which is:

[figure: ∀wφ[v]; ∀wφ[v]; ψ[y_{0}, z]]

It is constructively justified that one may search through that derivation to find the particular witnesses w_{1}, …, w_{n} that are needed to reach the quantifier free conclusion ψ. It is clear that in general we need finitely many of such witnesses, as it is demonstrated by the example ψ :≡ φ(w_{1}) ∧ φ(w_{2}) with undecidable φ(w) (this is exactly a contraction). When searching upwards, the derivation may branch and hence each branch may contribute a number of witnesses. The induction rule may as well increase this number and may make it depend on parameters. However, in the end we need only finitely many witnesses and the idea is to collect them all and form the conjunction of all these witnesses. Formally, we use a restricted universal quantifier and choose X (the number of witnesses w_{i} we need) and W (a functional with W i := w_{i} for i < X). This justifies the passage from (7) to

∃W X[(∀x<X) φ(v, W x) → ψ(y_{0}, z)].

Putting all together, we have reached ∀v∃y∀z∃W X[(∀x<X) φ(v, W x) → ψ(y, z)] and with an axiom of choice (which is justified by the Brouwer-Heyting-Kolmogorov-interpretation) we put the quantifiers in the right order and obtain the ^{∧}-translation of an implication:

∃W XY ∀vz[(∀x<Xvz) φ(v, W vzx) → ψ(Y v, z)].

We see that we do not only have a different translation but also the system T has changed: we now have an additional restricted universal quantifier. The new system is called T_{∧} and is T plus axioms for the restricted universal quantifier. Note that we do not increase the stock of functionals, but T_{∧} is no longer a quantifier free term calculus.

Diller and Nahm’s main result is: HA^{ω} is ^{∧}-interpretable in T_{∧}. With the new translation it is easy to interpret the contraction φ → φ ∧ φ (again we assume φ ≡ φ^{∧} ≡ ∀wφ_{0}):

(φ → φ ∧ φ)^{∧} ≡ ∃W X∀yz(∀x<X)φ_{0}(W x) → φ_{0}(y) ∧ φ_{0}(z)].

Now one just has to put X := 2 and W 0 := y and W 1 := z.

Diller and Vogel in [DV75] give an ^{∧}-interpretation of HA^{ω+} + BI_{σ} in T enriched by barrecursion functionals for type σ. Avigad in [Av98b] also uses a Diller-Nahm-style translation to interpret ÎD_{<ω} in Martin-Löf type theory, again a system with undecidable formulae.

We already mentioned that together with the negative translation, Gödel’s Dialectica interpretation is also applicable to classical PA. A more direct approach was proposed by Shoenfield (see [Sh67]). He gives a translation φ ↦ φ^{S} ≡ ∀x∃yφ_{S}(x, y). φ is said to be ^{S}- interpretable in T if there are terms T with T ⊢ φ_{S}(x, t), where in contrast to the Dialecticainterpretation the terms t may contain the free variables x. Shoenfield’s interpretation theorem is: PA is ^{S}-interpretable in T (to be more precise: Shoenfield gives an interpretation in the full type structure, but it is easy to see that this interpretation can be performed in T as well).

The two variants ^{∧} and ^{S} of the Dialectica translation were introduced to interpret two extensions of HA, namely HA^{ω} and PA. Both extensions have specific features that made a refined translation necessary: In the first case the lack of characteristic terms, in the second the law of excluded middle. Although the two problems are quite different, we can say that they are both of logical nature. In both cases we do not need new functionals for the interpretation (note: in terms of proof theoretic strength, PA, HA and HA^{ω} are equivalent), but a more careful translation that respects the peculiarities of the respective extensions. This turns out to be a general pattern: The translation is responsible for handling logic, while the functionals are needed to interpret the mathematics (here we cheat a bit: of course combinators and elementary arithmetic are also needed to handle logic, we mean the mathematically strong functionals R_{σ}).

3.b Functional interpretations of set theory

Having successfully applied functional interpretations to first-order arithmetic, the question arises if this method is applicable also to other theories, namely to set theory. The starting point of this work was the paper by Michael Rathjen ‘A proof-theoretic characterization of the primitive recursive set functions’ ([Ra92]). Rathjen was able to show that the Σ_{1}-definable set functions of Kripke-Platek set theory with infinity (KPω) but foundation restricted to Σ_{1}-formulae are exactly the primitive recursive set functions^{7} enriched by the constant function x ↦ ω. This result is closely related to a theorem by Parsons, namely that the provably recursive functions of IΣ_{1} are exactly the primitive recursive functions. The Shoenfield interpretation of PA with primitive recursive functionals can be considered as a generalization of Parsons’ theorem. It was a natural question to ask if it is also possible to raise Rathjen’s result to KPω in a similar way.

Andreas Weiermann suggested to generalize the primitive recursive set functions to functionals of finite type and he proposed to try a Shoenfield-style interpretation of KPω. This led to a system of set functionals with constants 0, ω; combinators K, S; case-distinction functionals C_{σ} for each type; a functional Suc with axiom Suc ab = a ∪ {b}; a union functional U with Ufa = ⋃{fx | x^{∈}a} and transfinite recursors R with RGa = G(RG ↾ a)a. Recall the general pattern from the end of section 3.a. It turned out (point two of the general pattern) that these functionals are sufficient to deal with the mathematics of KPω. Point one of this pattern now demands a suitable translation to handle the logic of KPω. Here the first attempt was to copy the Shoenfield-translation, but leaving Δ_{0}-formulae unchanged (this is necessary in order to interpret (Δ_{0}-Separation)). This approach leads to a problem we call Δ_{0} ⊆ Π_{1}, i.e., a restricted quantifier might be read as an unrestricted one as well. This is the crucial example

7This notion is due to Jensen and Carp, [JK71]

(8) ⊢ ∀w(w^{∈}a → ⊥) → (∀y^{∈}a)⊥.

The ^{S}-translation of this formula is (modulo classical logic)

∃w[w^{∈}a ∨ (∀y^{∈}a)⊥].

Thus an interpreting term for (8) corresponds to a choice functional. In a joint paper with Volker Hartung [BH98] we add such a choice functional to our system of functionals and can prove the interpretation theorem^{8}. However, this solution is not satisfactory: the choicefunctional is not definable within KPω. What we did is the following: we tried to repair a logical problem (Δ_{0} ⊆ Π_{1} is of a logical nature) by use of a new functional - this is not in the sense of our general pattern. Instead, we need a new translation to solve this problem.

Now we have reached the starting point of the first part of this thesis: Functional interpretations of KPω, Aczel’s constructive set theory CZF and related systems.

The first result we were able to establish was a Diller-Nahm-style interpretation of KPω in the full type structure over a model of KPω ([Bu97]). Later it turned out that this translation is the Shoenfield-variant of the translation ^{×} that we introduce for the interpretation of constructive set theory. Therefore we will now focus on the interpretation of constructive set theory.

The theory CZF is formulated in the first order language of set theory L_{∈} with ∈ as the only non-logical symbol. The logic of CZF is intuitionistic predicate logic with equality. CZF^{−} contains the following non-logical axioms: (Extensionality), (Pair), (Union), (Infinity), (Foundation), (Δ_{0}-Separation) and a strong version of the collection schema. CZF contains in additions a subset collection schema. An important feature of Aczel’s constructive set theory is a kind of predicativity: we do not have power sets and full separation, and the proof-theoretic strength of CZF is that of KPω. The ∈-relation is not decidable in CZF, and from a constructive point of view, this is an essential property that should be preserved by the mathematical axioms: Using ω and (Δ_{0}-Separation), we have arithmetical comprehension in CZF, i.e., a := {x^{∈}ω | φ(x)} is a set for all arithmetical formulae φ. Hence, decidability of ∈ would imply the law of excluded middle for arithmetical formulae, and CZF would be far from what one would call constructively justified. It is straightforward to see that ∅^{∈}x∨¬∅^{∈}x implies the excluded middle for all restricted formulae (restricted excluded middle (REM)). Since (Subset-Collection) implies the exponentiation axiom and this, together with (REM), implies the power set axiom (and therefore would increase the strength of CZF enormously), we see once more that non-decidability of ∈ is an important feature. Hence, care should be taken in formulating the axioms: The axiom of choice (AC) has to be excluded, since together with (Δ_{0}-Separation) it implies (REM) (this result is due to Dianonescu, cf. [Be85] p. 163). This fact is remarkable: In the context of arithmetic the Brouwer-Heyting-Kolmogorov interpretation verifies an axiom of choice while in constructive set theory (AC) has non-constructive consequences. These considerations show that we have to be careful when we are looking for functionals for our interpretation. All functionals that decide ∈ cannot be considered to be constructive, in particular not a choice functional, because the interpreting system should not be stronger than CZF itself.

8In that paper we do not fix an interpreting system but argue in the full type structure over a model of KPω

In [Ac78] Aczel gives his interpretation of CZF in ML_{1}V, i.e., Martin-Löf type theory with one universe and the type V of iterated sets, see also [Ac82], [GR94] and [TvD88] chapter 11, section 8. For more information about CZF see for example [Ac78].

Now we focus on the functional interpretation of CZF^{−}. The interpretation of all of CZF is similar and only needs an additional functional.

According to our general pattern, two steps are necessary towards a functional interpretation of constructive set theory. We look for a suitable translation to interpret the logic of CZF^{−} and for a system of functionals to treat the mathematics of CZF^{−}.

Towards the right translation we observe that constructive set theory contains undecidable prime formulae. In order to cope with the resulting contraction problem, we have to choose a Diller-Nahm style translation. Furthermore, Δ_{0}-formulae should be preserved by the translation since we have to interpret (Δ_{0}-Separation). Consequently, a first attempt is to define a translation that leaves Δ_{0}-formulae unchanged and then copies the Diller-Nahm translation. Doing so, we are able to interpret contraction and solve the Δ_{0} ⊆ Π_{1}-problem: The Diller-Nahm-translation of ∀w(w^{∈}a → ⊥) → (∀y^{∈}a)⊥ reads

∃W X[(∀x^{∈}X)(W x^{∈}a → ⊥) → (∀y^{∈}a)⊥]

and this is trivially interpretable by the definition X := a and W := λx.x. The contraction and the Δ_{0} ⊆ Π_{1}-problem are of similar nature and they are both solved by the Diller-Nahm translation: we accumulate information instead of facing an impossible decision. In the first case we collect two witnesses y and z, in the second a-many witnesses w for all w^{∈}a. The Diller-Nahm translation is designed exactly for that purpose.

However, within intuitionistic logic we also have to deal with the restricted existential quantifier and here we face a dual phenomenon, we call it the Δ_{0} ⊆ Σ_{1}-problem

⊢ (∃y^{∈}a)(y = y) → ∃w(w^{∈}a → w = w).

The translation of this formula is

(9) ∃w[(∃y^{∈}a)(y = y) → w^{∈}a].

Again any interpreting term for (9) involves a choice-functional, which is not constructive. A closer look at (9) yields that we are not able to give a single interpreting term, while it is very easy to give a set of interpreting terms, namely the set a = {x | x^{∈}a} itself. We would again like to accumulate information. This time, however, the Diller-Nahm translation does not help us at all. It only allows us to collect information in the antecedent of an implication. This observation leads to the main new idea behind the translation ^{×}: Its existential quantifiers do no longer range over single objects, but over inhabited sets in the sense of the Diller-Nahm translation. For every φ ∈ L_{∈} the translation φ^{×} reads

(10) ∃vQ∀w[(∃q^{∈}Q)(q = q) ∧ (∀q^{∈}Q)φ̄(vq, w)].

That means the existential quantifiers now range over inhabited sets {vq | q^{∈}Q}, where each element vq (q^{∈}Q) satisfies the matrix φ̄.

The second step towards a functional interpretation of CZF^{−} is to find a suitable system of constructive set functionals to interpret the mathematics of CZF^{−}. This job will be done by T_{∈}, the set theoretic analogue of Diller and Nahm’s system T_{∧}. The stock of basic functionals of T_{∈} is 0 and ω for the corresponding sets, combinators K and S, a successorfunctional Suc, I and N for intersection and relative complements, the union functional U and recursors R.

A distinguished and important subclass of the formulae of T_{∈} is Δ_{0}. It contains atomic formulae s^{∈}t, s = t and ⊥, where s and t are terms of type o and is closed under →, ∧, ∨, (∀x^{∈}a) and (∃x^{∈}a). Hence this class contains all Δ_{0}-formulae of L_{∈}. It is an important feature of Δ_{0}, that is does not contain equations of higher type. In order to axiomatize the basic functionals, T_{∈} contains furthermore equations of arbitrary type. Completely analogous to T_{∧} we now obtain the class of formulae of T_{∈} by closing under ∧, → and (∀x^{∈}a). Hence, the restricted existential quantifier and disjunctions occur only within Δ_{0}-formulae, while ∧, → and (∀x^{∈}a) may as well occur in front of equations of arbitrary type. This together with the fact that every Δ_{0}-formula is equivalent in T_{∈} to an equation of type o (see 4.8) underline the correspondence between T_{∈} and T_{∧}.

The main result of the first part of this thesis is: CZF^{−} is ^{×}-interpretable in T_{∈}. We want to give some hints of the proof of the interpretation theorem: Most of the logical rules and axioms are interpreted similarly to the Diller-Nahm interpretation since due to the distribution lemma 4.22 we can usually instantiate the extra existential quantifiers in (10) by singletons {0}.

Let us now turn to the interpretation of the mathematical axioms of CZF^{−}. We see that (Extensionality) is preserved by the translation and is also an axiom of T_{∈}. (Pair), (Union) and (Δ_{0}-Separation) are interpreted using Suc, I, N, U and infinity, of course, needs the constant ω. (Foundation) is interpreted by transfinite recursion (corresponding to arithmetic, where induction is interpreted by primitive recursion) - here we refer to the comments given within the proof in section 4. The interpretation of (Strong-Collection) uses the union functional U. It should be noted that we can interpret collection for arbitrary formulae. In contrast, in the classical framework, we can treat collection only for Σ_{1}-formulae. A similar phenomenon arises in the context of arithmetic, see also the comments in chapter 8.

There are two immediate corollaries of the interpretation theorem: CZF^{−} is consistent relative to T_{∈} and its provably total Σ_{1}-definable set functions are all representable in T_{∈} by terms of T_{∈}.

When it comes to all of CZF it is obvious that we need a further functional to interpret (Subset-Collection). Therefore we introduce a so-called fullness-functional^{9} and obtain the system T_{∈}^{+}. Now we easily extend the interpretation theorem to CZF. Vice versa we show that all extensions of T_{∈} that interpret CZF already contain this fullness-functional. Hence T_{∈}^{+} is the minimal extension of T_{∈} that interprets all of CZF.

Once having defined the^{×}-translation, it is a natural question to ask for a characterization of this translation. For this purpose we introduce constructive set theories in all finite types, i.e., we extend the language of T_{∈} to full first order logic by adding quantifiers for all finite types and define several extensions of T_{∈} formulated in this language. One of these extensions is CZF^{ω−}: it is the system ‘spanned’ by T_{∈} and CZF^{−} in a natural way. This theory is an adequate framework to discuss the strength of the translation ^{×}. Similar to the characterization of the Dialectica and the Diller-Nahm translation we ask which principles are needed to prove the schema φ ↔ φ^{×}. In arithmetic and for the Dialectica-translation, these principles are the axiom of choice

9note that (Subset-Collection) is equivalent to a fullness axiom

(AC) ∀x^{σ}∃y^{τ}φ(x, y) → ∃Y^{σ→τ}∀xφ(x, Y x)

and, furthermore, Markov’s principle (M) and a principle of independence of premises (IP). For the Diller-Nahm translation one needs (AC) and modified principles (M_{∧}) and (IP_{∧}). In both cases we also have a converse result, namely that the schemes φ ↔ φ^{D} (resp. φ ↔ φ^{∧}) already prove the principles mentioned above. As already mentioned, it is well known that the (set theoretical) axiom of choice added to CZF^{−} implies the law of excluded middle for restricted formulae (REM). Therefore, in intuitionistic set theory choice principles should be treated with great caution. However, we need certain choice principles to prove the equivalence of ∀uφ and φ → ψ to their ^{×}-translation. A closer look yields that the involved principles are weaker than those stated above. Here by ‘weaker’ we mean the following: Suppose that ∀x∃yφ holds. (AC) as above gives a choice functional Y with ∀xφ(x, Y x). In the context of set theory, we are in general not able to give such a functional, but only functionals S and Y such that for any argument x the set Sx is inhabited and φ(x, Y sx) holds for all s^{∈}Sx. This leads to the scheme (WAC) (weak axiom of choice)

(WAC) ∀x∃yφ(x, y) → ∃Y S ∀x[(∃s^{∈}Sx)(s = s) ∧ (∀s^{∈}Sx)φ(x, Y sx)].

Similar considerations lead to a variant of (IP_{∧}) which is appropriate in the context of set theory

(IP_{∈}) (∀wφ → ∃uψ) → ∃US[∀wφ → (∃s^{∈}S)(s = s) ∧ (∀s^{∈}S)ψ(Us)], φ ∈ T_{∈}.

The variant of Markov’s principle introduced by Diller and Nahm is sufficient also in our case:

(M_{∈}) (∀wφ → ψ) → ∃W X[(∀x^{∈}X)φ(W x) → ψ] φ, ψ ∈ T_{∈}.

We introduce a theory CZF_{∗}^{ω−} by adding these three principles to CZF^{ω−}. Our characterization theorem states the equivalence of CZF_{∗}^{ω−} and CZF^{ω−} + {φ ↔ φ^{×} | φ ∈ L_{∈}^{ω}}. We further show that CZF_{∗}^{ω−} is interpretable in T_{∈} and hence is a conservative extension of T_{∈}. We show that any formula interpretable in T_{∈} is derivable in CZF_{∗}^{ω−} and give characterizations of definable sets and set functions of CZF_{∗}^{ω−} in terms of functionals of T_{∈}. The results of this section also hold if we consider the extensions with (Subset-Collection) and add the fullness functional to T_{∈}.

Let us briefly return to Kripke-Platek set theory. We already mentioned that KPω contains the Δ_{0} ⊆ Π_{1} problem which in the presence of classical logic coincides with the Δ_{0} ⊆ Σ_{1}-problem. If we compare the Shoenfield versions of ^{∧} and ^{×}, we observe the same phenomenon: the differences collapse, and we find a single translation ^{∨} that is the common Shoenfield-version both of ^{×} and ^{∧}. If we consider the Shoenfield interpretation of PA, it turns out that the interpreting system is still Gödel’s T, i.e., a system with intuitionistic logic. This holds, since the part of T that is relevant for the interpretation of HA is decidable. With T_{∈} this is different and hence we need to introduce the classical counterpart T_{∈}^{c}. A first result is now: KPω is^{∨}-interpretable in T_{∈}^{c}. Again we can easily extend this to Kripke-Platek set theory in all finite types. We introduce the natural span of T_{∈}^{c} and KPω. We further characterize the translation ^{∨} (here the crucial axiom a principle related to (WAC), but only for formulae of T_{∈}^{c}). Now the interpretation theorem also holds for the extended systems and we obtain relative consistency, conservativity and a characterization of the provably total Σ-definable set functions. In [BH98] we also give a computability proof for the functionals of T_{∈}^{c} in KPω and hence we even have the stronger result: The provably total Σ-definable set functions of KPω^{ω}_{∗} are exactly the type 1 functionals of T_{∈}^{c}. By introducing appropriate new functionals it is straightforward to extend the interpretation to KPi, KPM and KPω + (Pow).

Functional interpretation of Aczel’s constructive set theory

4.a Systems T_{∈} and T_{∈}^{+} of constructive set functionals

In this section we define and investigate systems T_{∈} and T_{∈}^{+} of constructive set functionals of finite type. These systems are set theoretic counterparts to Diller and Nahm’s system T_{∧}.

Definition 4.1 (The System T_{∈}) T_{∈} contains the following basic terms (by a : σ we denote that a is a term of type σ): - countably many variables x^{σ}, y^{σ}, ⋯: σ for each type σ

- constants 0, ω : o

- combinators K_{τσ} : τ → σ → τ

- combinators S_{ρστ} : (ρ → σ → τ) → (ρ → σ) → ρ → τ

- Suc, I : o → o → o,

- N : o → o → o → o

- U : 1 → o → o

- R_{σ} : ((o → σ) → o → σ) → o → σ. The terms of T_{∈} are defined inductively:

- Each basic term of type σ is a term of type σ.

- if a : σ → τ and b : σ then (ab) : τ.

Again, Δ_{0} is the least class of formulae that contains atomic formulae s^{∈}t, s = t, ⊥ and is closed under ∧, ∨, →, (∀x^{∈}s) and (∃x^{∈}s), where s and t are terms of type o, x : o and x ∉ F V (s). Note that Δ_{0} does not contain equations of higher types but may contain terms of arbitrary type as sub-terms.

The formulae of T_{∈} are defined inductively:

- Δ_{0}-formulae are formulae of T_{∈}.

- s = t is an atomic formula if s, t are terms of the same type σ. We call this an equation of type σ.

- if φ, ψ are formulae of T_{∈}, then (φ → ψ), (φ ∧ ψ) are formulae of T_{∈}.

- if φ is a formula of T_{∈} and s : o does not contain the variable x : o, then (∀x^{∈}s)φ is a formula of T_{∈}.

Remark: Note that formally the restricted quantifiers are abbreviations. An important property of the formulae of T_{∈} is, that ∨ and (∃x^{∈}s) do not occur in front of equations of higher type. With corollary 4.8 Δ_{0}-formulae are equivalent to equations - this underlines that T_{∈} is the set theoretic counterpart of T_{∧}. Terms without variables are called functionals. The combinators K and S allow a definition of λx^{σ}.a^{τ} : σ → τ for all terms a : τ:

- λx^{σ}.x^{σ} :≡ S_{σ(o→σ)σ}K_{σ(o→σ)}K_{σo}
- λx^{σ}.a^{τ} :≡ K_{τ→σ}a^{τ} if x^{σ} does not occur in a^{τ}.
- λx^{σ}.a^{σ→τ}x^{σ} :≡ a^{σ→τ} if x^{σ} does not occur in a^{τ}.
- λx^{σ}.a^{ρ→τ}b^{ρ} :≡ S_{σ→ρ→τ}(λx^{σ}.a)(λx^{σ}.b) else.

We now give the axioms and rules of T_{∈}. The defining axiom for the recursor R_{σ} needs a restriction functional ↾ which is definable by terms of T_{∈} (for the definition see 4.9). In lemma 4.9 we’ll see that T_{∈} proves all the properties of the restriction functionals that are needed in order to guarantee that R_{σ} is well-defined. We assume that all terms have suitable types.

4.a Systems T_{∈} and T_{∈}^{+} of constructive set functionals  45

(i) positive sentential logic A1 − A11, the equality-axiom (for equality of arbitrary type) (ii) rules and axioms Q1 − Q4 for restricted quantifiers (iii) (Foundation): (∀u^{∈}a)φ(u) → φ(a) ⊢ φ(b) a ∉ F V (φ) (iv) (Set-Extensionality): (∀z^{∈}a)z^{∈}b ∧ (∀z^{∈}b)z^{∈}a → a = b

(v) ¬x^{∈}0 (vi) x^{∈}ω ↔ x = 0 ∨ (∃y^{∈}ω)x = Suc yy (vii) x^{∈} Suc ab ↔ x^{∈}a ∨ x = b i.e. Suc ab = a ∪ {b} (viii) x^{∈}Iab ↔ x^{∈}b ∧ (∀y^{∈}a)x^{∈}y i.e. Iab = b ∩ ⋂ a (ix) x^{∈}Nabc ↔ x^{∈}a ∧ (x^{∈}b → x^{∈}c) i.e. Nabc = {x^{∈}a | x^{∈}b → x^{∈}c} (x) x^{∈}Ufa ↔ (∃y^{∈}a)x^{∈}fy i.e. Ufa = ⋃{fx | x^{∈}a} (xi) RGa = G[(RG)↾a]a (xii) Kab = a (xiii) Sabc = ac(bc) (xiv) (Ext-rule): φ → sa = ta ⊢ φ → s = t a ∉ F V (φ, s, t). a is eigenvariable of (Foundation) and of the (Ext-rule). Concerning equality in higher types, our system is related to WE − HA^{ω} in the terminology of [Tr73], see also the remark 4.19 and section 2.e.4. We now examine the expressive power of the system T_{∈}.

Lemma 4.2 (The λ-operator) T_{∈} proves the following properties of the λ-operator:

(i) (λx^{σ}.a^{τ}) b^{σ} = a^{τ}[x^{σ} := b] (ii) b = c → λx.(a[y := b]) = λx.(a[y := c]) if x is not free in b and c. □

Lemma 4.3 (ξ-rule) The ξ-rule is admissible in T_{∈}:

φ → s = t ⊢ φ → λz.s = λz.t, if z is not free in φ. Proof: From the premise it follows with 4.2 (i)

φ → (λz.s)z = (λz.t)z. Hence an application of (Ext-Rule) yields the conclusion. □

Definition 4.4 (Zero-functionals) We define canonical zero-functionals 0^{σ} for each type σ:

(i) 0^{o} :≡ 0 (ii) 0^{ρ→τ} :≡ λx^{ρ}.0^{τ} Similarly, we define F^{o} :≡ F and F^{ρ→τ} :≡ λx^{ρ}.F^{τ} for each functional F : o.

Lemma 4.5 (Terms for set operations) The following operations on sets are definable by terms of T_{∈}:

{a}, ⋃ a, {a, b}, a ∪ b, a − b, a ∩ b, ⋃{t(x) | x^{∈}s}, {t(x) | x^{∈}s}. We can further define numerals \underline{n} for n ∈ ℕ. Usually we write n for \underline{n}. Proof: {a} = Suc 0a, ⋃ a = U(λx.x)a, {a, b} = Suc{a}b, a ∪ b = ⋃{a, b}, a − b = Nab0, a ∩ b = I{a}b, this is since x^{∈}I{a}b ↔ x^{∈}b ∧ (∀y^{∈}{a})x^{∈}y ↔ x^{∈}b ∧ x^{∈}a, ⋃\{t(x) | x^{∈}s\} = U(λx.t(x))s, \{t(x) | x^{∈}s\} = ⋃\{\{t(x)\} | x^{∈}s\}, \underline{0} :≡ 0, \underline{n + 1} :≡ Suc \underline{n} \underline{n}. □ By definition we have:

Lemma 4.6 T_{∈} ⊢ (∀x^{∈}1)x = 0 ∧ (∀x^{∈}2)(x = 0 ∨ x = 1) ∧ (∀x^{∈}ω)(x = 0 ∨ (∃y^{∈}x)(y = y)). □

Lemma 4.7 (Separation terms) For every formula φ(x⃗, z) ∈ Δ_{0} there is a term t_{φ}(x⃗, a) such that T_{∈} ⊢ w^{∈}t_{φ}(x⃗, a) ↔ w^{∈}a ∧ φ(x⃗, w).

4.a Systems T_{∈} and T_{∈}^{+} of constructive set functionals  47

Note that we do not have separation terms for arbitrary formulae of T_{∈}, in particular not for formulae containing equations of higher type. Proof: Using set extensionality is suffices to show the assertion for φ that does not contain =. The proof is by induction on the build-up of φ:

(i) φ is atomic. 1. φ ≡ ⊥. Then define t_{φ} :≡ 0.

2. φ ≡ s^{∈}r, where s, r : o may contain x⃗, w. Define

t_{φ} :≡ ⋃{{w | z^{∈}{s} ∩ r} | w^{∈}a}

which we abbreviate by {w | z^{∈}{s} ∩ r, w^{∈}a}.^{1} This definition yields:

w^{∈}t_{φ} ↔ w^{∈}a ∧ s^{∈}r.

(ii) φ ≡ φ_{0} ∧ φ_{1} and φ ≡ φ_{0} ∨ φ_{1}. These cases follow immediately by the induction hypothesis using ∩ and ∪.

(iii) φ ≡ φ_{0} → φ_{1}. By I.H. we have terms t_{φ_{0}}, t_{φ_{1}}. Define t_{φ} :≡ Nat_{φ_{0}}t_{φ_{1}}:

w^{∈}Nat_{φ_{0}}t_{φ_{1}} ↔ w^{∈}a ∧ (w^{∈}t_{φ_{0}} → w^{∈}t_{φ_{1}}) ↔ w^{∈}a ∧ φ.

(iv) φ ≡ (∃x^{∈}s) φ_{0}(x⃗, x, w) where s may contain x⃗, w. The induction hypothesis provides a term t_{φ_{0}}(x⃗, x, a) with w^{∈}t_{φ_{0}} ↔ w^{∈}a ∧ φ_{0}(x⃗, x, w). We define (recall the notation used in (i))

t_{φ} :≡ {w | z^{∈}{w} ∩ t_{φ_{0}}, x^{∈}s, w^{∈}a}.

Then we have

w^{∈}t_{φ} ↔ w^{∈}a ∧ (∃x^{∈}s) w^{∈}t_{φ_{0}} ↔_{I.H.} w^{∈}a ∧ (∃x^{∈}s) (w^{∈}a ∧ φ_{0}(x⃗, x, w)) ↔
w^{∈}a ∧ (∃x^{∈}s) φ_{0}(x⃗, x, w).

(v) φ ≡ (∀x^{∈}s) φ_{0}(x⃗, x, w). The induction hypothesis provides t_{φ_{0}}(x⃗, x, a) as above. First we define a term

t̃(w) :≡ I{{w | z^{∈}{w} ∩ t_{φ_{0}}} | x^{∈}s}a.

1By lemma 4.5 we may define a term {t(z) | z^{∈}{s} ∩ r}. For t(z) :≡ (λz.w)z this reads {w | z^{∈}{s} ∩ r} and is a term that contains the free parameter w.

This yields by induction hypothesis

y^{∈}t̃(w) ↔ y^{∈}a ∧ (∀x^{∈}s) y^{∈}{w | z^{∈}{w} ∩ t_{φ_{0}}} ↔ y^{∈}a ∧ (∀x^{∈}s)[y = w ∧ w^{∈}t_{φ_{0}}]
↔_{I.H.} y^{∈}a ∧ (∀x^{∈}s)[y = w ∧ w^{∈}a ∧ φ_{0}(x⃗, x, w)] ↔ w^{∈}a ∧ (∀x^{∈}s) φ_{0}(x⃗, x, w).

Hence t_{φ} :≡ {w | z^{∈}{w} ∩ t̃(w), w^{∈}a} yields:

w^{∈}t_{φ} ↔ w^{∈}a ∧ w^{∈}t̃(w) ↔ w^{∈}a ∧ (∀x^{∈}s) φ_{0}(x⃗, x, w).

This completes the proof of the lemma. □

Corollary 4.8 (Characteristic terms for Δ_{0}-formulae) For every φ ∈ Δ_{0} there is a term χ_{φ} with

T_{∈} ⊢ φ ↔ χ_{φ} = 1.

Hence every formula φ of type o is equivalent to an equation of type o.

Proof: The previous lemma yields φ ↔ 0^{∈}1 ∧ φ ↔ 0^{∈}t_{φ} ↔ {0} ∩ t_{φ} = {0}. Hence we may define χ_{φ} := {0} ∩ t_{φ}. □

Lemma 4.9 (Restriction functionals) There are functionals ↾_{τ} of type (o → τ) → o → o → τ such that (we write f↾a for ↾fa):

(i) T_{∈} ⊢ x^{∈}a → (f↾a)x = fx

(ii) T_{∈} ⊢ (∀x^{∈}a)(gx = fx) → (f↾a) = (g↾a).

These two properties guarantee that the recursors R_{σ} are well-defined. Proof: By induction on τ: 1. τ = o. Define

f↾a :≡ λx. ⋃{fz | z^{∈}({x} ∩ a)}.

Ad (i): For x^{∈}a we have f↾ax = ⋃{fz | z^{∈}({x} ∩ a)} = ⋃{fz | z^{∈}{x}} = ⋃{fx} = fx. Ad (ii): We choose a fresh variable y and have

(∀x^{∈}a)fx = gx → f↾ay = ⋃{fz | z^{∈}({y} ∩ a)} = ⋃{gz | z^{∈}({y} ∩ a)} = g↾ay

and by an application of the (Ext-Rule)

(∀x^{∈}a)fx = gx → f↾a = g↾a.

2. τ ≡ σ → ρ. Suppose f : o → σ → ρ. Then we define

f↾a :≡ λx^{o}.λz^{σ}.[((λy^{o}.fyz)↾a)x]

with z a fresh variable of type σ. The properties (i) and (ii) follow easily using the induction hypothesis and the ξ-rule. □

Lemma 4.10 (Case distinction functionals, definition by cases) 1. There are terms C_{σ} : σ → σ → o → o → σ in T_{∈} such that

(1) T_{∈} ⊢ a^{∈}b → Cxyab = x ∧ ¬a^{∈}b → Cxyab = y.

2. For terms t_{0}, t_{1} of type τ there is a term t of type o → τ such that

(2) T_{∈} ⊢ x = 0 → tx = t_{0} ∧ (∃y^{∈}x)y = y → tx = t_{1}.

Note that we neither have T_{∈} ⊢ x = 0 ∨ (∃y^{∈}x)y = y nor T_{∈} ⊢ a^{∈}b ∨ ¬a^{∈}b. Proof: 1. By induction on σ. If σ = o define

C_{o} :≡ λxyab. ⋃({x | a^{∈}b} ∪ {y | ¬a^{∈}b}).

Then we obtain (1): a^{∈}b → C_{o}xyab = ⋃{x} = x and ¬a^{∈}b → C_{o}xyab = ⋃{y} = y. If σ ≡ ρ → τ we take fresh variables x : ρ → τ, y : ρ → τ, z : ρ and define

C_{σ} :≡ λxyab.(λz.C_{τ}(xz)(yz)ab).

2. By induction on τ. 2.1. τ = o. We define

t :≡ λx. ⋃{t_{0} | z^{∈}{x} ∩ 1} ∪ ⋃{t_{1} | z^{∈}x}.

Then we have t0 = ⋃{t_{0} | z^{∈}1} ∪ {t_{1} | z^{∈}0} = t_{0}. If (∃y^{∈}x)y = y we have tx = t_{1}. 2.2 τ ≡ σ → ρ is similar to the case above using λ-abstraction. □

Lemma 4.11 (Ordered pairs, projections) We define the following functionals (), ()_{0}, ()_{1} of type o → o → o, o → o and o → o resp.:

(i) (x, y) :≡ {{x}, {x, y}}

(ii) (x)_{0} :≡ ⋃{y | y ∈ ⋃ x ∧ (∀z^{∈}x)y^{∈}z}

(iii) (x)_{1} :≡ ⋃{y | y ∈ ⋃ x ∧ (∃z^{∈}x)¬y^{∈}z}.

Then T_{∈} proves

((x_{0}, x_{1}))_{0} = ⋃{y | y^{∈}{x_{0}, x_{1}} ∧ (∀z^{∈}{{x_{0}}, {x_{0}, x_{1}}})y^{∈}z} = x_{0}

and similarly ((x_{0}, x_{1}))_{1} = x_{1}. Iterating this argument we can obviously define ordered ntuples and corresponding projections for each natural number n. Via λ-abstraction it is straightforward to define ordered pairs and projections also for objects of arbitrary finite type: For each τ there are terms ( , ), ( )_{0} and ( )_{1} with T_{∈} ⊢ ((x_{0}, x_{1}))_{0} = x_{0} ∧ ((x_{0}, x_{1}))_{1} = x_{1}.□

This lemma gives:

Lemma 4.12 (Simultaneous recursion) Let n ⩾ 1. For terms s_{i} : (o → σ_{0}) → … → (o → σ_{n}) → o → σ_{i} (i = 0, …, n) there are corresponding terms t_{0} ∈ o → σ_{0}, …, t_{n} ∈ o → σ_{n} such that the following scheme of simultaneous recursion holds:

T_{∈} ⊢ t_{i}a = s_{i}(t_{0}↾a) … (t_{n}↾a)a

for i = 0, …, n and a ∈ o. □

Lemma 4.13 (Transitive closure) There is a type-1 functional TC with

(1) T_{∈} ⊢ (a ⊆ TC a) ∧ tran(TC a) ∧ (tran(b) ∧ a ⊆ b → TC a ⊆ b),

where tran(x) :≡ (∀y^{∈}x)(∀z^{∈}y)(z^{∈}x).

Proof: Let G :≡ λf.λx. ⋃{fz | z^{∈}x} ∪ x and TC :≡ RG. Then

TC a = ⋃{TC z | z^{∈}a} ∪ a.

By ∈-induction on a we prove (1). □

Lemma 4.14 (ω-recursion) Suppose that W is of type o → σ → σ and z of type σ. Then there is a functional F of type o → σ such that

T_{∈} ⊢ F 0 = z ∧ n^{∈}ω → F (n + 1) = W n (F n).

Proof: Using lemma 4.10 we define

Gfx = \{ \begin{matrix}z, & if x = 0 \\ W (⋃ x)(f ⋃ x), & if (∃y^{∈}x)y = y. \end{matrix}

For F :≡ RG we have

F 0 = RG0 = G(RG↾0)0 = z,

and since n^{∈}ω → ⋃(n + 1) = n

n^{∈}ω → F (n + 1) = RG(n + 1) = G(RG↾(n + 1))(n + 1) = W n(F n)

and we obtain the assertion. □

Lemma 4.15 (Finite sequences) By a finite sequence of elements of a given set a we mean a set f with

T_{∈} ⊢ fun(f) ∧ dom(f)^{∈}ω ∧ rng(f) ⊆ a.

We use the standard notations for finite sequences and write ⟨⟩ for the empty sequence, lh(s) for dom(s), s(n) for the n-th entry of s and s^{⌢}⟨x⟩ for s ∪ (lh(s), x). T_{∈} allows us to define the set of all finite sequences of elements of a for any given set a. We call this set Finseq a.

Proof: By ω-recursion we define

T 0 :≡ {0}, T (n + 1) :≡ {f ∪ {(n, x)} | f^{∈}T n, x^{∈}a}

and let Finseq a :≡ ⋃{T n | n^{∈}ω}. □ The following lemma will be useful in the proof of the interpretability of the foundation rule.

Lemma 4.16 Suppose z : σ, w : σ → o → σ. Then there is F : o → σ definable in T_{∈} such that

(1) T_{∈} ⊢ F ⟨⟩ = z ∧ [fun(s) ∧ dom(s)^{∈}ω → F (s^{⌢}⟨x⟩) = w(F s)x].

Proof: By ω-recursion we define

F_{0}s0 :≡ z, F_{0}s(n + 1) :≡ w(F_{0}sn)(s(n))

and F s :≡ F_{0}s(lh s). Then F fulfills (1). □

It is sure that T_{∈} does not contain an exponentiation functional that, given a, b, computes the set of all functions from a to b. Therefore (cf. corollary 4.29); it seems to be necessary to add a new functional to obtain a system that is strong enough to interpret (Subset-Collection). In view of lemma 2.2 we add a fullness functional:

Definition 4.17 (The extension T_{∈}^{+}) We extend the system T_{∈} by an new constant F : o → o → o and the following defining axiom

(F) R^{∈} mv(^{a}b) → (∃d^{∈}F ab)R : a >−< d.

We call the extended system T_{∈}^{+}.

Lemma 4.18 (Deduction theorem for T_{∈} and T_{∈}^{+}) T_{∈} and T_{∈}^{+} satisfy the deduction theorem. More precisely:

T_{∈}^{(+)} + φ ⊢ ψ ⇔ T_{∈}^{(+)} ⊢ φ → ψ,

provided that in “⇒” no free variable of φ is the eigenvariable of an application of a rule in the derivation T_{∈}^{(+)} + φ ⊢ ψ.

Proof: The non-trivial direction is “⇒”. The proof by induction on derivations in T_{∈}^{(+)} + φ is easy. We mention one case, namely (Foundation). Suppose the last inference was

T_{∈}^{(+)} + φ ⊢ (∀u^{∈}a)χ(u) → χ(a) ⇒ T_{∈}^{(+)} + φ ⊢ χ(b)

By induction hypothesis we obtain

T_{∈}^{(+)} ⊢ φ → (∀u^{∈}a)χ(u) → χ(a),

and whence

T_{∈}^{(+)} ⊢ (∀u^{∈}a)(φ → χ(u)) → (φ → χ(a)).

(Foundation) yields (due to the assumption a is not free in φ and hence φ[a := b] ≡ φ)

T_{∈}^{(+)} ⊢ φ → χ(b).

The remaining rules with eigenvariable are similar, all other cases straightforward. □

Remark 4.19 (Equality in higher types, the (Ext-Rule)) The systems T_{∈} and T_{∈}^{+} contain =^{σ} for each σ as primitives. Equality at the ground type is axiomatized by the usual set theoretical extensionality axiom a = b ↔ (∀x^{∈}a)(x^{∈}b) ∧ (∀x^{∈}b)(x^{∈}a). For higher types the situation is different. Here we include the (Ext-Rule) (which implies the ξ-rule) and hence equality for higher types is an extensional equality. We want to emphasize that this was made for technical requirements: Consider for example the restriction functionals: For type o we define

(∗) f ↾_{o} a :≡ λx. ⋃{fz | z^{∈}({x} ∩ a)}.

However, the verification of property (ii) for ↾_{o} needs the (Ext-rule). When it comes to ↾_{σ} for arbitrary σ we make the definition by pushing the restriction functional down to the ground-level, then using (∗) and finally raising it up to σ via λ-abstraction. The verification of the demanded properties again needs the (Ext-Rule). In the same way we define case distinction functionals in 4.10. Ordered pairs and projections for higher types and hence simultaneous recursion also uses this technique. We should mention, however, that in this case we could instead add product types σ × τ and corresponding functionals to T_{∈}.

Recall 2.13. In sections 4.a and 4.b we further use the following conventions:

Convention 4.20 Though it is an abuse of notation, we write in the sequel vq :≡ v_{1}q_{1}, …, v_{n}q_{n} for v_{i} ∈ o → σ_{i}, q_{i} ∈ o, v ≡ v_{1}, …, v_{n}, q ≡ q_{1}, …, q_{n}. In the next section we assign to any formula φ ∈ L_{∈} a formula φ^{×} of the following shape:

∃v_{1} … v_{n}Q_{1} … Q_{n}∀w_{1} … w_{m}((∃q^{∈}Q_{1})(q = q) ∧ … ∧ (∃q^{∈}Q_{n})(q = q)∧
(∀q_{1}^{∈}Q_{1}) … (∀q_{n}^{∈}Q_{n}) φ̄(v_{1}q_{1}, …, v_{n}q_{n}, w_{1}, …, w_{m}))

We will abbreviate a formula of that shape by

∃vQ ∀w (∀q^{∈}Q^{≠∅}) φ̄(vq, w),

where implicitly we assume the following conditions to be fulfilled:

- v and Q are (possibly empty) tuples of the same length, v_{i} : o → σ_{i}, Q_{i} : o and σ_{i} are arbitrary finite types for i = 1, …, n

- (∀q^{∈}Q^{≠∅}) φ̄(vq, w) ≡ (∃q^{∈}Q_{1})(q = q) ∧ .. ∧ (∃q^{∈}Q_{n})(q = q) ∧ (∀q_{1}^{∈}Q_{1})..(∀q_{n}^{∈}Q_{n})φ̄(v_{1}q_{1}, .., v_{n}q_{n}, w_{1}, .., w_{m})

- w is a (possibly empty) tuple of arbitrary finite length and with components of arbitrary finite type

- φ̄ a formula of T_{∈}, Q ∉ F V (φ̄) and F V (φ) = F V (φ^{×}).

Note that (∀s^{∈}S≠∅) φ is only an abbreviation for (∃s^{∈}S)(s = s) ∧ (∀s^{∈}S)φ (i.e. S ≠ ∅ abbreviates that S is inhabited). The following notation will be convenient: Suppose that X, S ≡ S_{1}, …, S_{n} are all of type o and W_{i} : o → o → σ_{i} for i = 1, …, m. Then one might define

X′ :≡ {(x, s_{1}, …, s_{n}) | x^{∈}X, s_{1}^{∈}S_{1}, …, s_{n}^{∈}S_{n}} and W′_{i}x :≡ W_{i}(x)_{i}(x)_{0}

where ()_{i} denotes the i-th projection for i = 0, …, n. For this definition we’ll just write

X′ :≡ {(x, s) | x^{∈}X, s^{∈}S} and W′x :≡ W (x)_{1}(x)_{0}.

If R_{i} : o, S_{i} : o; T_{i} : o, Y_{i} : o → τ_{i} for i = 1, …, n one might define for all i = 1, …, n:

R′_{i} :≡ {(r, s) | r^{∈}R_{i}, s^{∈}S_{i}}, Y′_{i}r :≡ Y_{i}(r)_{0}(r)_{1}.

In this case we use the notation

R′ :≡ {(r, s) | r^{∈}R; s^{∈}S}, Y′r :≡ (r)_{0}(r)_{1}.

Finally we abbreviate ⋃{{t(x) | x^{∈}X(y)} | y^{∈}Y } by {t(x) | x^{∈}X(y), y^{∈}Y }.

4.b Functional interpretation of CZF^{−} and CZF

This section is the heart of chapter 4. We define the translation ^{×} and give functional interpretations of CZF^{−} and CZF .

Definition 4.21 (The translation^{×}) We assign a formula φ^{×} to every formula φ ∈ L_{∈}, where φ^{×} has the following shape:

∃v_{1} … v_{n}Q_{1} … Q_{n}∀w_{1} … w_{m}((∃q^{∈}Q_{1})(q = q) ∧ … ∧ (∃q^{∈}Q_{n})(q = q)∧
(∀q_{1}^{∈}Q_{1}) … (∀q_{n}^{∈}Q_{n})φ̄(v_{1}q_{1}, …, v_{n}q_{n}, w_{1}, …, w_{m}))

with φ̄ a formula of T_{∈}. We will abbreviate a formula of that shape by (for the precise conventions see 4.20)

∃vQ ∀w (∀q^{∈}Q^{≠∅}) φ̄(vq, w).

1. φ^{×} :≡ φ for φ ∈ Δ_{0}. For the translation of formulae not in Δ_{0} now assume that

φ^{×} ≡ ∃vQ ∀w (∀q^{∈}Q^{≠∅}) φ̄(vq, w) and ψ^{×} ≡ ∃yR ∀z (∀r^{∈}R^{≠∅}) ψ̄(yr, z)

4.b Functional interpretation of CZF^{−} and CZF  55

are already defined with disjoint sequences of variables v_{i} : o → σ_{i}, w_{i} : τ_{i}, y_{i} : o → ρ_{i}, z_{i} : π_{i}.

2. (φ ∧ ψ)^{×} :≡ ∃vyQR∀wz(∀q^{∈}Q^{≠∅}) (∀r^{∈}R^{≠∅}) (φ̄ ∧ ψ̄)
3. (φ ∨ ψ)^{×} :≡ ∃vyT QRS∀wz

(∀q^{∈}Q^{≠∅}) (∀r^{∈}R^{≠∅}) (∀s^{∈}S≠∅) [T s^{∈}2 ∧ (T s = 0 → φ̄) ∧ (T s = 1 → ψ̄)]

4. (∃uφ)^{×} :≡ ∃UvSQ∀w(∀s^{∈}S≠∅) (∀q^{∈}Q^{≠∅}) φ̄(vq, w, Us)
5. (∀uφ)^{×} :≡ ∃V QT S∀wu(∀t^{∈}T^{≠∅}) (∀s^{∈}S^{≠∅}) (∀q^{∈}Qus≠∅) φ̄(V uqt, w, u)
6. (φ → ψ)^{×} :≡ ∃W XY RS∀vQz(∀s^{∈}S^{≠∅})

[(∀x^{∈}XvQzs_{2})(∀q^{∈}Q^{≠∅}) φ̄(vq, W vQzs_{1}x) → (∀r^{∈}RvQs_{4}^{≠∅}) ψ̄(Y vQrs_{3}, z)] if w is not the empty tuple. For w empty, we define:

7. (φ → ψ)^{×} :≡ ∃Y RS∀vQz(∀s^{∈}S^{≠∅})

[(∀q^{∈}Q^{≠∅}) φ̄(vq) → (∀r^{∈}RvQs_{4}^{≠∅}) ψ̄(Y vQrs_{3}, z)],

where

in 3: T : 1, S : o

in 4: U : 1, S : o

in 5: V_{i} : o → o → o → σ_{i}, Q_{i} : o → o → o, T_{i} : o, S_{i} : o for i = 1, …, n

in 6. X : (o → σ) → o → π → o → o a tuple of length 1, S a tuple of length lh(w)+lh(X)+ lh(y)+lh(R) and for s^{∈}S let s ≡ s_{1}s_{2}s_{3}s_{4} with lh(s_{1}) = lh(w), …, lh(s_{4}) = lh(R). The components of the tuples W, Y, R and S have the following types: W_{i} : (o → σ) → o → π → o → o → τ_{i}, Y_{i} : (o → σ) → o → o → o → ρ_{i}, R_{i} : (o → σ) → o → o → o, S_{i} : o.

in 7: Similar conditions as in 6.

This completes the definition of the translation ^{×}. A formula φ ∈ L_{∈} is called ^{×}-interpretable in T_{∈}, if there are terms v, Q in T_{∈} containing at most the free variables of φ such that

T_{∈} ⊢ (∀q^{∈}Q^{≠∅}) φ̄(vq, w).

Remark: Obviously we have ψ(t)^{×} :≡ ψ(x)^{×}[x := t].

Lemma 4.22 (Distribution lemma) Suppose that (φ → ψ)^{×} is as in definition 4.21, 6..

(i) If W, X, Y, R, S are interpreting terms for φ → ψ, then we find terms W′, X′, Y′, R′ depending on S of appropriate type such that

(1) (∀x^{∈}X′vQz)(∀q^{∈}Q^{≠∅}) φ̄(vq, W′vQzx) → (∀r^{∈}R′vQ^{≠∅}) ψ̄(Y′vQr, z).

(ii) If there are terms W′, X′, Y′, R′ with (1), then φ → ψ is ^{×}-interpretable in T_{∈}.

Proof: (i) The idea of the proof is the following: For every tuple s^{∈}S by hypothesis we have an implication

(2) (∀x^{∈}XvQzs_{2})(∀q^{∈}Q^{≠∅}) φ̄(vq, W vQzs_{1}x) → (∀r^{∈}RvQs_{4}^{≠∅}) ψ̄(Y vQrs_{3}, z)

Using effective unions we ‘glue’ all these implications together and obtain one single implication. This technique of effective unions will be an important tool later in the proof of the interpretation theorem. For this lemma define (note the conventions made in 4.20)

X′vQz :≡ {(s_{1}, x) | x^{∈}XvQzs_{2}, s_{1}^{∈}S_{1}, s_{2}^{∈}S_{2}} and W′vQzx :≡ W vQz(x)_{0}(x)_{1}
R′vQ :≡ {(s_{3}, r) | r^{∈}RvQs_{4}, s_{3}^{∈}S_{3}, s_{4}^{∈}S_{4}} and Y′vQr :≡ Y vQ(r)_{0}(r)_{1}

Suppose the premise of (1) holds:

(3) (∀x^{∈}X′vQz)(∀q^{∈}Q^{≠∅}) φ̄(vq, W′vQzx)

and show

(∀r^{∈}R′vQ^{≠∅}) ψ̄(Y′vQr, z).

(a) We have to show that R′vQ is inhabited. Since φ → ψ is interpretable we find s_{1}, s_{2}, s_{3}, s_{4}^{∈}S_{1}, S_{2}, S_{3}, S_{4}. By (3) and the definition of X′ and W′ we have

(∀x^{∈}XvQzs_{2})(∀q^{∈}Q^{≠∅}) φ̄(vq, W vQzs_{1}x),

hence with (2) we find an r^{∈}RvQs_{4}. Then (s_{3}, r)^{∈}R′vQ.

(b) It remains to show

(∀r^{∈}R′vQ)ψ̄(Y′vQr, z).

Take arbitrary r_{0}^{∈}R′vQ, i.e., r_{0} = (s_{3}, r) and choose s_{1}^{∈}S_{1}, s_{2}^{∈}S_{2} and s_{4}^{∈}S_{4} with r^{∈}RvQs_{4}.

(3) yields

(∀x^{∈}XvQzs_{2})(∀q^{∈}Q^{≠∅}) φ̄(vq, W vQzs_{1}x)

hence with (2) we obtain

(∀r^{∈}RvQs_{4}^{≠∅}) ψ̄(Y vQrs_{3}, z)

and therefore

ψ̄(Y′vQ(s_{3}, r), z).

(ii) Define S :≡ 1 (i.e., S_{i} = 1 and s_{i} = 0 for all s_{i}^{∈}S_{i} and all i = 1, …, n) and XvQzs_{2} :≡ X′vQz, W vQzs_{1}x :≡ W′vQzx, Y vQrs_{3} :≡ Y′vQr and RvQs_{4} :≡ R′vQ. □

Now we are able to prove the interpretation theorem:

Theorem 4.23 (Interpretation theorem) CZF^{−} is ^{×}-interpretable in T_{∈}.

Proof: By induction on derivations of CZF^{−}. We argue informally in T_{∈}. To simplify the notation we use the translations already given in definition 4.21.

A1: φ → φ. Define W_{0}v_{0}Q_{0}w_{1}x :≡ w_{1}, Xv_{0}Q_{0}w_{1} :≡ 1, V_{1}v_{0}Q_{0}q :≡ v_{0}q, Q_{1}v_{0}Q_{0} :≡ Q_{0}. This yields

(∀x^{∈}Xv_{0}Q_{0}w_{1})(∀q^{∈}Q_{0}^{≠∅}) φ̄(v_{0}q, W_{0}v_{0}Q_{0}w_{1}x) → (∀q^{∈}Q_{1}v_{0}Q_{0}^{≠∅}) φ̄(V_{1}v_{0}Q_{0}q, w_{1})

and hence by distribution lemma follows the interpretability of A1.

A2: φ, φ → ψ ⊢ ψ. By induction hypothesis and distribution lemma we have v_{0}, Q_{0} such that

(1) (∀q^{∈}Q_{0}^{≠∅}) φ̄(v_{0}q, w_{0})

and W_{1}, X_{1}, Y_{1}, R_{1} such that

(∀x^{∈}X_{1}vQz)(∀q^{∈}Q^{≠∅}) φ̄(vq, W_{1}vQzx) → (∀r^{∈}R_{1}vQ^{≠∅}) ψ̄(Y_{1}vQr, z)

hence by substituting v_{0} for v and Q_{0} for Q

(2) (∀x^{∈}X_{1}v_{0}Q_{0}z)(∀q^{∈}Q_{0}^{≠∅}) φ̄(v_{0}q, W_{1}v_{0}Q_{0}zx) → (∀r^{∈}R_{1}v_{0}Q_{0}^{≠∅}) ψ̄(Y_{1}v_{0}Q_{0}r, z).

(1) yields

(3) (∀x^{∈}X_{1}v_{0}Q_{0}z)(∀q^{∈}Q_{0}^{≠∅}) φ̄(v_{0}q, W_{1}v_{0}Q_{0}zx)

and hence an application of A2 in T_{∈} to (2) and (3) gives

(∀r^{∈}R_{1}v_{0}Q_{0}^{≠∅}) ψ̄(Y_{1}v_{0}Q_{0}r, z).

Now we get interpreting terms y :≡ Y_{1}v_{0}Q_{0} and R :≡ R_{1}v_{0}Q_{0} for ψ.

A3: φ → ψ, ψ → χ ⊢ φ → χ. By induction hypothesis and distribution lemma we have W_{i}, X_{i}, Y_{i}, R_{i} (i = 1, 2) such that

(∀x_{1}^{∈}X_{1}v_{1}Q_{1}z_{1})(∀q_{1}^{∈}Q_{1}^{≠∅}) φ̄(v_{1}q_{1}, W_{1}v_{1}Q_{1}z_{1}x_{1}) → (∀r_{1}^{∈}R_{1}v_{1}Q_{1}^{≠∅}) ψ̄(Y_{1}v_{1}Q_{1}r_{1}, z_{1})

(∀x_{2}^{∈}X_{2}v_{2}Q_{2}z_{2})(∀q_{2}^{∈}Q_{2}^{≠∅}) ψ̄(v_{2}q_{2}, W_{2}v_{2}Q_{2}z_{2}x_{2}) → (∀r_{2}^{∈}R_{2}v_{2}Q_{2}^{≠∅}) χ̄(Y_{2}v_{2}Q_{2}r_{2}, z_{2}).

We are looking for W, X, Y, R such that

(1) (∀x^{∈}XvQz)(∀q^{∈}Q^{≠∅}) φ̄(vq, W vQzx) → (∀r^{∈}RvQ^{≠∅}) χ̄(Y vQr, z).

We substitute z for z_{2},v for v_{1}, Q for Q_{1}, Y_{1}vQ for v_{2}, t :≡ W_{2}(Y_{1}vQ)(R_{1}vQ)zx_{2} for z_{1}, R_{1}vQ for Q_{2}, define s :≡ X_{2}(Y_{1}vQ)(R_{1}vQ)z and obtain

(∀x_{2}^{∈}s)(∀x_{1}^{∈}X_{1}vQt)(∀q^{∈}Q^{≠∅}) φ̄(vq, W_{1}vQtx_{1}) → (∀x_{2}^{∈}s)(∀r_{1}^{∈}R_{1}vQ^{≠∅}) ψ̄(Y_{1}vQr_{1}, t)

(∀x_{2}^{∈}s)(∀r_{1}^{∈}R_{1}vQ^{≠∅}) ψ̄(Y_{1}vQr_{1}, t) → (∀r_{2}^{∈}R_{2}(Y_{1}vQ)(R_{1}vQ)^{≠∅}) χ̄(Y_{2}(Y_{1}vQ)(R_{1}vQ)r_{2}, z)

A3 yields

(∀x_{2}^{∈}s)(∀x_{1}^{∈}X_{1}vQt)(∀q^{∈}Q^{≠∅}) φ̄(vq, W_{1}vQtx_{1}) →
(∀r_{2}^{∈}R_{2}(Y_{1}vQ)(R_{1}vQ)^{≠∅}) χ̄(Y_{2}(Y_{1}vQ)(R_{1}vQ)r_{2}, z).

Defining

XvQz :≡ {(x_{1}, x_{2}) | x_{1}^{∈}X_{1}vQt, x_{2}^{∈}s}, W vQzx :≡ W_{1}vQ(W_{2}(Y_{1}vQ)(R_{1}vQ)z(x)_{1})(x)_{0}

RvQ :≡ R_{2}(Y_{1}vQ)(R_{1}vQ), Y vQ :≡ Y_{2}(Y_{1}vQ)(R_{1}vQ)

we fulfill (1).

A4: (φ ∧ ψ) → φ: Define Xv_{0}Q_{0}y_{0}R_{0}w_{1} :≡ 1, W_{0}v_{0}Q_{0}y_{0}R_{0}w_{1}x :≡ w_{1}, Z_{0}v_{0}Q_{0}y_{0}R_{0}w_{1}x :≡ 0^{π}, V_{1}v_{0}Q_{0}y_{0}R_{0} :≡ v_{0}, Q_{1}v_{0}Q_{0}y_{0}R_{0} :≡ Q_{0}. This interprets A4:

(∀x^{∈}Xv_{0}Q_{0}y_{0}R_{0}w_{1})(∀q^{∈}Q_{0}^{≠∅}) (∀r^{∈}R_{0}^{≠∅}) [φ̄(v_{0}q, W_{0}v_{0}Q_{0}y_{0}R_{0}w_{1}x)∧

ψ̄(y_{0}r, Z_{0}v_{0}Q_{0}y_{0}R_{0}w_{1}x)] → (∀q^{∈}Q_{1}v_{0}Q_{0}y_{0}R_{0}^{≠∅}) φ̄(V_{1}v_{0}Q_{0}y_{0}R_{0}q, w_{1}).

A5/A6: φ_{0} → φ_{1} → φ_{2} ⊢ φ_{0} ∧ φ_{1} → φ_{2} and φ_{0} ∧ φ_{1} → φ_{2} ⊢ φ_{0} → φ_{1} → φ_{2}: φ_{0} → φ_{1} → φ_{2} is interpretable if and only if there are terms W_{0}, X_{0}, W_{1}, X_{1}, V_{2}, Q_{2} such that

(1) (∀x_{0}^{∈}X_{0}v_{0}Q_{0}v_{1}Q_{1}w_{2})(∀q_{0}^{∈}Q_{0}^{≠∅}) φ̄_{0}(v_{0}q_{0}, W_{0}v_{0}Q_{0}v_{1}Q_{1}w_{2}x_{0}) →
(∀x_{1}^{∈}X_{1}v_{0}Q_{0}v_{1}Q_{1}w_{2})(∀q_{1}^{∈}Q_{1}^{≠∅}) φ̄_{1}(v_{1}q_{1}, W_{1}v_{0}Q_{0}v_{1}Q_{1}w_{2}x_{1}) →
(∀q_{2}^{∈}Q_{2}v_{0}Q_{0}v_{1}Q_{1}^{≠∅}) φ̄_{2}(V_{2}v_{0}Q_{0}v_{1}Q_{1}q_{2}, w_{2}).

φ_{0} ∧ φ_{1} → φ_{2} is interpretable if and only if there are terms W_{0}, W_{1}, X, V_{2}, Q_{2} such that

(2) (∀x^{∈}Xv_{0}Q_{0}v_{1}Q_{1}w_{2})(∀q_{0}^{∈}Q_{0}^{≠∅}) (∀q_{1}^{∈}Q_{1}^{≠∅}) [φ̄_{0}(v_{0}q_{0}, W_{0}v_{0}Q_{0}v_{1}Q_{1}w_{2}x)∧
φ̄_{1}(v_{1}q_{1}, W_{1}v_{0}Q_{0}v_{1}Q_{1}w_{2}x)] → (∀q_{2}^{∈}Q_{2}v_{0}Q_{0}v_{1}Q_{1}^{≠∅}) φ̄_{2}(V_{2}v_{0}Q_{0}v_{1}Q_{1}q_{2}, w_{2}).

Suppose there are W_{0}, X_{0}, W_{1}, X_{1}, V_{2}, Q_{2} with (1). Then X :≡ X_{1} ∪ X_{2} yields (2). Suppose vice versa that W_{0}, W_{1}, X, V_{2}, Q_{2} fulfill (2). Putting X_{0} :≡ X, X_{1} :≡ X we obtain (1). A7: φ → ψ, φ → χ ⊢ φ → ψ ∧ χ This rule contains a contraction and now we benefit from the fact that^{×} possesses all features of the Diller-Nahm translation. By induction hypothesis and distribution lemma we have W_{i}, X_{i}, Y_{i}, R_{i} such that

(∀x_{1}^{∈}X_{1}v_{1}Q_{1}z_{1})(∀q_{1}^{∈}Q_{1}^{≠∅}) φ̄(v_{1}q_{1}, W_{1}v_{1}Q_{1}z_{1}x_{1}) → (∀r_{1}^{∈}R_{1}v_{1}Q_{1}^{≠∅}) ψ̄(Y_{1}v_{1}Q_{1}r_{1}, z_{1})

(∀x_{2}^{∈}X_{2}v_{2}Q_{2}z_{2})(∀q_{2}^{∈}Q_{2}^{≠∅}) φ̄(v_{2}q_{2}, W_{2}v_{2}Q_{2}z_{2}x_{2}) → (∀r_{2}^{∈}R_{2}v_{2}Q_{2}^{≠∅}) χ̄(Y_{2}v_{2}Q_{2}r_{2}, z_{2})

We are looking for W, X, Y′_{i}, R′_{i} such that

(∀x^{∈}XvQz_{1}z_{2})(∀q^{∈}Q^{≠∅}) φ̄(vq, W vQz_{1}z_{2}x) →
(∀r_{1}^{∈}R′_{1}vQ^{≠∅}) (∀r_{2}^{∈}R′_{2}vQ^{≠∅}) (ψ̄(Y′_{1}vQr_{1}, z_{1}) ∧ χ̄(Y′_{2}vQr_{2}, z_{2}))

We get this by the following definitions:

XvQz_{1}z_{2} :≡ {(x, 0) | x^{∈}X_{1}vQz_{1}} ∪ {(x, 1) | x^{∈}X_{2}vQz_{2}},

W vQz_{1}z_{2}x :≡ \{ \begin{matrix}W_{1}vQz_{1}(x)_{0}, & (x)_{1} = 0 \\ W_{2}vQz_{2}(x)_{0}, & (x)_{1} = 1 \end{matrix} (cf. lemma 4.10)

and further R′_{i}, Y′_{i} :≡ R_{i}, Y_{i}.

A8: φ → φ ∨ ψ is easy using Zero-functionals (cf. definition 4.4): Define Xv_{0}Q_{0}w_{1}z_{1} :≡ 1, W_{0}v_{0}Q_{0}w_{1}z_{1}x :≡ w_{1}, Q_{1}v_{0}Q_{0} :≡ Q_{0}, V_{1}v_{0}Q_{0}q :≡ v_{0}q, Rv_{0}Q_{0} :≡ 1, Y v_{0}Q_{0}r :≡ 0^{ρ}, Sv_{0}Q_{0} :≡ 1 and T v_{0}Q_{0}s :≡ 0. Then we have

(∀x^{∈}Xv_{0}Q_{0}w_{1}z_{1})(∀q^{∈}Q_{0}^{≠∅}) φ̄(v_{0}q, W_{0}v_{0}Q_{0}w_{1}z_{1}x) → (∀q^{∈}Q_{1}v_{0}Q_{0}^{≠∅}) (∀r^{∈}Rv_{0}Q_{0}^{≠∅})

(∀s^{∈}Sv_{0}Q_{0}≠∅) ((T v_{0}Q_{0}s^{∈}2) ∧ (T v_{0}Q_{0}s = 0 → φ̄(v_{0}q, w_{1})) ∧ (T v_{0}Q_{0}s = 1 → ψ̄(0, z_{1})))

A9: ψ → φ ∨ ψ is similar.

A10: φ → χ, ψ → χ ⊢ φ ∨ ψ → χ Case 1: Both φ and ψ are in Δ_{0}. By induction hypothesis and distribution lemma we have R_{i}, Y_{i} such that

φ → (∀r_{1}^{∈}R_{1}^{≠∅}) χ̄(Y_{1}r_{1}, z) ψ → (∀r_{2}^{∈}R_{2}^{≠∅}) χ̄(Y_{2}r_{2}, z).

We need terms R, Y such that

φ ∨ ψ → (∀r^{∈}R^{≠∅}) χ̄(Y r, z).

Therefore we define

R :≡ \{(r, 0) | r^{∈}R_{1} ∧ φ\} ∪ \{(r, 1) | r^{∈}R_{2} ∧ ψ\} and Y r :≡ \{ \begin{matrix}Y_{1}(r)_{0}, & if (r)_{1} = 0 \\ Y_{2}(r)_{0}, & if (r)_{1} = 1. \end{matrix}

Case 2: Not both φ and ψ are in Δ_{0}. By hypothesis we have W_{i}, X_{i}, Y_{i}, R_{i} such that

(1) (∀x_{1}^{∈}X_{1}v_{1}Q_{1}z_{1})(∀q_{1}^{∈}Q_{1}^{≠∅}) φ̄(v_{1}q_{1}, W_{1}v_{1}Q_{1}z_{1}x_{1}) → (∀r_{1}^{∈}R_{1}v_{1}Q_{1}^{≠∅}) χ̄(Y_{1}v_{1}Q_{1}r_{1}, z_{1})

(2) (∀x_{2}^{∈}X_{2}v_{2}Q_{2}z_{2})(∀q_{2}^{∈}Q_{2}^{≠∅}) ψ̄(v_{2}q_{2}, W_{2}v_{2}Q_{2}z_{2}x_{2}) → (∀r_{2}^{∈}R_{2}v_{2}Q_{2}^{≠∅}) χ̄(Y_{2}v_{2}Q_{2}r_{2}, z_{2})

We are looking for W′_{i}, X, Y, R such that

(3) (∀x^{∈}Xv_{1}v_{2}T Q_{1}Q_{2}Sz)(∀q_{1}^{∈}Q_{1}^{≠∅}) (∀q_{2}^{∈}Q_{2}^{≠∅}) (∀s^{∈}S≠∅) [T s^{∈}2∧
(T s = 0 → φ̄(v_{1}q, W_{1}v_{1}v_{2}T Q_{1}Q_{2}Szx)) ∧ (T s = 1 → ψ̄(v_{2}q, W_{2}v_{1}v_{2}T Q_{1}Q_{2}Szx))]

→ (∀r^{∈}Rv_{1}v_{2}T Q_{1}Q_{2}S≠∅) χ̄(Y v_{1}v_{2}T Q_{1}Q_{2}Sr, z)

We define (we need to add {0} to X for similar reasons as in Q4, see the motivation given there):

Xv_{1}v_{2}T Q_{1}Q_{2}Sz :≡ X_{1}v_{1}Q_{1}z ∪ X_{2}v_{2}Q_{2}z ∪ {0}

W′_{1}v_{1}v_{2}T Q_{1}Q_{2}Szx :≡ W_{1}v_{1}Q_{1}zx W′_{2}v_{1}v_{2}T Q_{1}Q_{2}Szx :≡ W_{2}v_{2}Q_{2}zx

Rv_{1}v_{2}T Q_{1}Q_{2}S :≡

{(r, 0) | r^{∈}R_{1}v_{1}Q_{1} ∧ (∃s^{∈}S)(T s = 0)} ∪ {(r, 1) | r^{∈}R_{2}v_{2}Q_{2} ∧ (∃s^{∈}S)(T s = 1)}

Y v_{1}v_{2}T Q_{1}Q_{2}r :≡ \{ \begin{matrix}Y_{1}v_{1}Q_{1}(r)_{0}, & if (r)_{1} = 0 \\ Y_{2}v_{2}Q_{2}(r)_{0}, & if (r)_{1} = 1. \end{matrix}

Now suppose the antecedent of (3) holds. Since 0^{∈}X we see that S is inhabited, say s_{0}^{∈}S. By assumption we have T s_{0} = 0 or T s_{0} = 1 (cf. lemma 4.6), w.l.o.g. T s_{0} = 0. By definition the antecedent of (1) holds, hence its conclusion. This yields

(∀r^{∈}{(r, 0) | r^{∈}R_{1}v_{1}Q_{1} ∧ (∃s^{∈}S)(T s = 0)}≠∅) χ̄(Y v_{1}v_{2}T Q_{1}Q_{2}Sr, z).

If there further exits s^{∈}S with T s = 1 a similar argument shows

(∀r^{∈}{(r, 1) | r^{∈}R_{2}v_{2}Q_{2} ∧ (∃s^{∈}S)(T s = 1)}≠∅) χ̄(Y v_{1}v_{2}T Q_{1}Q_{2}Sr, z)

and we are done.

A11 : ⊥ → φ is interpretable by A11.

The interpretation of the identity rule is easy.

Q1: ∀uφ(u) → φ(a) It suffices to find terms W_{0}, U_{0}, X, V_{1}, Q_{1} such that

(∀x^{∈}XV_{0}Q_{0}ST w_{1})(∀s^{∈}S^{≠∅}) (∀t^{∈}T^{≠∅}) (∀q^{∈}Q_{0}(U_{0}V_{0}Q_{0}ST w_{1}x)s≠∅)
φ̄(V_{0}(U_{0}V_{0}Q_{0}ST w_{1}x)qt, W_{0}V_{0}Q_{0}ST w_{1}x, U_{0}V_{0}Q_{0}ST w_{1}x) →

(∀q^{∈}Q_{1}V_{0}Q_{0}ST^{≠∅}) φ̄(V_{1}V_{0}Q_{0}ST q, w_{1}, a).

Define

U_{0}V_{0}Q_{0}ST w_{1}x :≡ a, W_{0}V_{0}Q_{0}ST w_{1}x :≡ w_{1}, XV_{0}Q_{0}ST w_{1} :≡ 1

Q_{1}V_{0}Q_{0}ST :≡ {(q, t) | q^{∈}Q_{0}as, t^{∈}T, s^{∈}S}, V_{1}V_{0}Q_{0}ST q :≡ V_{0}a(q)_{0}(q)_{1}.

Q2: φ(a) → ∃uφ(u)

It suffices to find terms W_{0}, X, U_{1}, V_{1}, S_{1}, Q_{1} such that

(∀x^{∈}Xv_{0}Q_{0}w_{1})(∀q^{∈}Q_{0}^{≠∅}) φ̄(v_{0}q, W_{0}v_{0}Q_{0}w_{1}x, a) →
(∀s^{∈}Sv_{0}Q_{0}≠∅) (∀q^{∈}Q_{1}v_{0}Q_{0}^{≠∅}) φ̄(V_{1}v_{0}Q_{0}q, w_{1}, U_{1}v_{0}Q_{0}s).

Define

Xv_{0}Q_{0}w_{1} :≡ 1, W_{0}v_{0}Q_{0}w_{1}x :≡ w_{1}, Sv_{0}Q_{0} :≡ 1, Q_{1}v_{0}Q_{0} :≡ Q_{0}, V_{1}v_{0}Q_{0} :≡ v_{0}, U_{1}v_{0}Q_{0}s :≡ a.

Q3: φ → ψ(u) ⊢ φ → ∀uψ(u).

Case 1: ∀uψ(u) is in Δ_{0}, i.e. ψ ≡ (u^{∈}a → ψ_{0}(u)) with ψ_{0} ∈ Δ_{0}. By induction hypothesis and distribution lemma we have W_{0}, X_{0}[u] (by W_{0}, X_{0}[u] we denote that both W_{0} and X_{0} may contain the free variable u) such that

(1) (∀x^{∈}X_{0}vq)(∀q^{∈}Q^{≠∅}) φ̄(vq, W_{0}vQx) → (u^{∈}a → ψ_{0}(u)).

We define

XvQ :≡ {(u, x) | x^{∈}X_{0}vQ, u^{∈}a}, W vQx :≡ W_{0}[u := (x)_{0}]vQ(x)_{1}.

This interprets the conclusion:

(∀x^{∈}XvQ)(∀q^{∈}Q^{≠∅}) φ̄(vQ, W vQx) → (∀u^{∈}a)ψ_{0}(u).

Case 2: ∀uψ(u) ∉ Δ_{0}. By induction hypothesis and distribution lemma we have terms W_{0}, X_{0}, Y_{0}, R_{0}[u] such that

(∀x^{∈}X_{0}vQz)(∀q^{∈}Q^{≠∅}) φ̄(vq, W_{0}vQzx) → (∀r^{∈}R_{0}vQ^{≠∅}) ψ̄(Y_{0}vQr, z, u)

Define XvQzu :≡ X_{0}vQz, W vQzux :≡ W_{0}vQx, SvQ, T vQ :≡ 1, RvQsu :≡ R_{0}vQ, Y vQtru :≡ Y_{0}vQr and obtain

(∀x^{∈}XvQzu)(∀q^{∈}Q^{≠∅}) φ̄(vq, W vQzux) → (∀s^{∈}1^{≠∅}) (∀t^{∈}1^{≠∅}) (∀r^{∈}RvQsu≠∅) ψ̄(Y vQtru, z, u)

Q4: φ(u) → ψ ⊢ ∃uφ(u) → ψ Case 1: ∃uφ(u) is in Δ_{0}, i.e. φ(u) ≡ (u^{∈}a ∧ φ_{0}(u)) with φ_{0} ∈ Δ_{0}. This is the crucial case which is responsible for the Δ_{0} ⊆ Σ_{1}-problem, see also the discussion in chapter 3. By induction hypothesis and distribution lemma we now have Y_{0}, R_{0}[u] such that

u^{∈}a ∧ φ_{0}(u) → (∀r^{∈}R_{0}^{≠∅}) ψ̄(Y_{0}r, z).

We define (note that we need separation terms for φ ∈ Δ_{0})

R :≡ {(u, r) | r^{∈}R_{0}, u^{∈}a ∧ φ_{0}(u)}, Y r :≡ Y_{0}[u := (r)_{0}](r)_{1}

This interprets the conclusion:

(∃u^{∈}a)φ_{0}(u) → (∀r^{∈}R^{≠∅}) ψ̄(Y r, z).

Case 2: ∃uφ(u) ∉ Δ_{0}. This case also needs some comments: The^{×}-translation transforms an existential quantifier ∃u into a pair of existential quantifiers ∃US. The existence of an u with φ(u) is equivalent to the existence of an inhabited set {Us | s^{∈}S} with φ(Us) for all s^{∈}S. In rule Q4 we derive a formula with an existential quantifier in the antecedent. Therefore via the translation the following might happen: while the premise of the rule provides a witness u, the pair U, S given by the conclusion does not entail any information because S is the empty set. We avoid this case by adding a single point 0 in the definition of the set X below to guarantee that S is always inhabited. By induction hypothesis and distribution lemma we have terms W_{0}, X_{0}, Y_{0}, R_{0}[u] such that

(1) (∀x^{∈}X_{0}vQz)(∀q^{∈}Q^{≠∅}) φ̄(vq, W_{0}vQzx, u) → (∀r^{∈}R_{0}vQ^{≠∅}) ψ̄(Y_{0}vQr, z).

We need terms X, W, Y, R such that

(2) (∀x^{∈}XUvSQz)(∀s^{∈}S≠∅) (∀q^{∈}Q^{≠∅}) φ̄(vq, W UvSQzx, Us) →
(∀r^{∈}RUvSQ≠∅) ψ̄(Y UvSQr, z).

We set

XUvSQz :≡ {(s, x) | x^{∈}X_{0}[u := Us]vQz, s^{∈}S} ∪ {0},
W UvSQzx :≡ W_{0}[u := U(x)_{0}]vQz(x)_{1}
RUvSQ :≡ {(s, r) | r^{∈}R_{0}[u := Us]vQ, s^{∈}S}, Y UvSQr :≡ Y_{0}[u := U(r)_{0}]vQ(r)_{1}.

Suppose that the premise of (2) holds and show

(∀r^{∈}RUvSQ^{≠∅}) ψ̄(Y UvSQr, z).

Since 0^{∈}X we find s_{0}^{∈}S by the premise of (2). Again using the premise of (2) we get

(∀x^{∈}X_{0}[u := Us_{0}]vQz)(∀q^{∈}Q^{≠∅}) φ̄(vq, W_{0}[u := Us_{0}]vQzx, Us_{0}).

Note that we have (∀s^{∈}S) in the premise of (2), hence we can use the information in particular for s = s_{0}. (1) now gives

(∀r^{∈}R_{0}[u := Us_{0}]vQ≠∅) ψ̄(Y_{0}[u := Us_{0}]vQr, z).

This argument holds for arbitrary s^{∈}S, therefore (2) is proved.

This completes the interpretation of the logic of CZF^{−}. It remains to show the interpretability of the set-theoretical axioms and rules: (Extensionality): This is a Δ_{0}-formula; it is unchanged by ^{×} and an axiom of T_{∈}.

(Pair), (Union): By lemma 4.5.

(Δ_{0}-Separation): By lemma 4.7.

(Infinity): By the constant ω.

(Strong-Collection): (∀x^{∈}a)∃wφ(x, w) → ∃zφ′(a, z), where φ′(a, z) denotes (∀x^{∈}a)(∃y^{∈}z)φ(x, y) ∧ (∀y^{∈}z)(∃x^{∈}a)φ(x, y). The translation of the collection schema is long and complex. For better readability we split the schema into two implications

(1) (∀x^{∈}a)∃wφ(x, w) → ∃z(∀x^{∈}a)(∃y^{∈}z)φ(x, y) and
(2) (∀x^{∈}a)∃wφ(x, w) → ∃z(∀y^{∈}z)(∃x^{∈}a)φ(x, y).

Since we’ll interpret z in both cases by the same term this yields the interpretability of the collection schema. We further omit the less important or superfluous parameters (those for example which are interpreted by {0}) and suppose that φ^{×} ≡ ∃vQ ∀w (∀q^{∈}Q^{≠∅}) φ̄(vq, w). We assume that we have already done the standard substitutions. For the interpretation of (1) we essentially need a term Z such that^{2}

[x^{∈}a → (∀q^{∈}Q^{≠∅}) (∀s^{∈}S≠∅) φ̄(vqx, w, x, Y sx)] →

[x^{∈}a → (∀q^{∈}Q^{≠∅}) (∀s^{∈}S≠∅) (Y sx^{∈}Z ∧ φ̄(vqx, w, x, Y sx))].

That means: we need a set Z that contains witnesses y_{x} satisfying φ̄(vq, w, x, y_{x}) for all x^{∈}a. Since for any x^{∈}a by the premise Y computes the witness y_{x} we define

Z :≡ {Y sx | x^{∈}a, s^{∈}S}.

To show that Z interprets also (2), we essentially use the advantage of the ^{×}-translation. Here we need functionals R and X such that for any y^{∈}Z = {Y sx | x^{∈}a, s^{∈}S} we have (∀r^{∈}Ry≠∅) φ̄(ṽq, w_{1}, Xr, y). We just define (suppressing some superfluous parameters)

Ry :≡ {x^{∈}a | (∃s^{∈}S)Y sx = y}, Xr :≡ r,
Q̃y :≡ {< q, x > | q^{∈}Qx, (∃s^{∈}Sx)y = Y sx, x^{∈}a} and ṽqy :≡ v(q)_{0}(q)_{1}

and interpret (2):

[x^{∈}a → (∀q^{∈}Qx^{≠∅}) (∀s^{∈}Sx≠∅) φ̄(vqx, w, x, Y sx)] →
[y^{∈}{Y sx | x^{∈}a, s^{∈}S} → (∀q^{∈}Q̃y^{≠∅}) (∀r^{∈}Ry≠∅) (r^{∈}a ∧ φ̄(ṽq, w, Xr, y))].

The situation here is similar to (Q4): We are only able to give a set R of interpreting terms for (2) and not a single term. Again the ^{×}-translation allows us the interpretation.

(Foundation): (∀u^{∈}a)φ(u) → φ(a) ⊢ φ(b) provided a ∉ φ(u). Suppose φ(a) ≡ ∃vQ∀w(∀q^{∈}Q^{≠∅}) φ̄(vq, w, a). By induction hypothesis, distribution lemma and several trivial substitutions with 1 we have terms W_{0}, U_{0}, X_{0}, V_{1}, Q_{1} (by λ-abstraction we consider a as an additional parameter, i.e., write W_{0}V_{0}Q_{0}wxa) such that:

(1) (∀x^{∈}X_{0}V_{0}Q_{0}wa)[U_{0}V_{0}Q_{0}wxa^{∈}a → (∀q^{∈}Q_{0}(U_{0}V_{0}Q_{0}wxa)≠∅) φ̄(V_{0}(U_{0}V_{0}Q_{0}wxa)q, W_{0}V_{0}Q_{0}wxa, U_{0}V_{0}Q_{0}wxa)] →

2To be more precise: according to the ^{×}-translation we need a pair Z, T, but T will be just 1 and hence we suppress it.

(∀q^{∈}Q_{1}V_{0}Q_{0}a≠∅) φ̄(V_{1}V_{0}Q_{0}aq, w, a).

By simultaneous recursion (cf. lemma 4.12) we define

Qa :≡ Q_{1}(v↾a)(Q↾a)a and va :≡ V_{1}(v↾a)(Q↾a)a.

We substitute in (1) v↾a for V_{0} and Q↾a for Q_{0} and obtain

(2) (∀x^{∈}X_{0}(vQ↾a)wa)[U_{0}(vQ↾a)wxa^{∈}a → (∀q^{∈}(Q↾a)(U_{0}(vQ↾a)wxa)≠∅)

φ̄(v↾a(U_{0}(vQ↾a)wxa)q, W_{0}(vQ↾a)wxa, U_{0}(vQ↾a)wxa)] →

(∀q^{∈}Qa≠∅) φ̄(vaq, w, a).

To interpret the conclusion of the foundation rule, we have to show for arbitrary w, b:

(3) (∀q^{∈}Qb^{≠∅}) φ̄(vbq, w, b).

The proof of (3) needs some comments. In arithmetic the interpretation of the induction rule uses essentially the function · −: that is, one counts backwards starting from b and defines explicitly all the parameters w_{i} that are needed to verify the conclusion. In set theory there is no operation · − on sets. Instead, we do the following: By a Skolem-process we define a set T that contains all needed parameters and is closed under the application of W_{0}. Since W_{0} might be of arbitrary type σ, T is a set of finite sequences coding finite sets of objects of type σ. We define, using lemma 4.13, 4.14, 4.15:

g :≡ TC({b} ∪ {X_{0}v↾uQ↾uwu | u^{∈} TC({b})}),

T :≡ Finseq(g × g) i.e. T :≡ {⟨(x_{1}, y_{1}), …, (x_{n}, y_{n})⟩ | x_{i}, y_{i}^{∈}g; n^{∈}ω} and

F ⟨⟩ :≡ w, F (s^{⌢}⟨(x, y)⟩) :≡ W_{0}(v↾y)(Q↾y)(F s)xy.

By ∈-induction on c we show ψ(c), where

ψ(c) :≡ c^{∈}g → (∀s^{∈}T)(∀q^{∈}Qc≠∅) φ̄(vcq, F s, c).

For c = b and s = ⟨⟩ this yields (3). Suppose that for all u^{∈}c we have ψ(u). To show ψ(c) we suppose c^{∈}g and choose s^{∈}T and have to show

(∀q^{∈}Qc≠∅) φ̄(vcq, F s, c).

By (2) it suffices to show that for all x^{∈}X_{0}(vQ↾c)wc and all u^{∈}c

(4) (∀q^{∈}Q↾cu≠∅) φ̄(v↾cuq, W_{0}(vQ↾c)(F s)xc, u)

holds. It is s^{⌢}⟨(x, c)⟩^{∈}T and hence with ψ(u) we get

(∀q^{∈}Qu≠∅) φ̄(vuq, F (s^{⌢}⟨(x, c)⟩), u).

The definition

F (s^{⌢}⟨(x, c)⟩) = W_{0}(v↾c)(Q↾c)(F s)xc

and vu = v↾cu, Qu = Q↾cu gives (4). □

Corollary 4.24 CZF^{−} is consistent relative to T_{∈}. □

As a further consequence of the interpretation theorem we show that all provably total Σ-definable set functions of CZF^{−} are represented in T_{∈} by terms of T_{∈}:

Corollary 4.25 Let φ(x, y) :≡ ∃zφ_{0}(x, y, z), φ_{0} ∈ Δ_{0} and CZF^{−} ⊢ ∃!yφ(x, y). Then there are functionals F, S, Z such that

T_{∈} ⊢ (∀s^{∈}Sx≠∅) φ_{0}(x, F x, Zsx).

Proof: The uniqueness part of the assumption implies

CZF^{−} ⊢ φ_{0}(x, y_{1}, z_{1}) ∧ φ_{0}(x, y_{2}, z_{2}) → y_{1} = y_{2},

this is a Δ_{0}-formula, hence by the interpretation theorem it is derivable also in T_{∈}. The interpretation theorem applied to CZF^{−} ⊢ ∃yzφ_{0}(x, y, z) provides terms Y, Z_{0}, Q, S_{0} with free parameter x and

T_{∈} ⊢ (∀q^{∈}Q≠∅) (∀s^{∈}S_{0}≠∅) φ_{0}(x, Y q, Z_{0}s).

Define

F x :≡ ⋃{Y q | q^{∈}Q}, S :≡ λx.S_{0} and Z :≡ λx.Z_{0}.

By T_{∈} ⊢ (∀q_{1}, q_{2}^{∈}Q)Y q_{1} = Y q_{2} we have T_{∈} ⊢ (∀q^{∈}Q)F x = Y q ∧ (∃q^{∈}Q)F x = Y q and hence

T_{∈} ⊢ q_{0}^{∈}Q → F x = ⋃{Y q | q^{∈}Q} = ⋃{Y q_{0}} = Y q_{0}.

This yields the assertion. □

In the rest of this section we interpret the theory CZF (which adds (Subset-Collection) to CZF^{−}) in the extension T_{∈}^{+} of T_{∈}. By lemma 2.2 CZF is equivalent to CZF^{−} plus the fullness axiom. Therefore the interpretation of CZF is a corollary of the following lemma:

Lemma 4.26 ∀ab∃c Full(c, a, b) is ^{×}-interpretable in T_{∈}^{+}.

Proof: Consider the translation

(∃c Full(c, a, b))^{×} ≡ ∃CQ∀R(∀q^{∈}Q≠∅) [R ∈ mv(^{a}b) → ∃d ∈ Cq R : a >−< d].

By putting Q :≡ 1 and C :≡ λx.F ab (the fullness functional) we get

T_{∈}^{+} ⊢ (∀q^{∈}Q≠∅) [R ∈ mv(^{a}b) → ∃d ∈ Cq R : a >−< d]. □

Corollary 4.27 CZF is ^{×}-interpretable in T_{∈}^{+}. □

It is no surprise that a fullness functional interprets (Subset-Collection). Vice-versa, we can show that any extension of T_{∈} that interprets all of CZF already contains such a fullness functional. Therefore T_{∈}^{+} is a minimal extension of T_{∈} that interprets CZF:

Theorem 4.28 Suppose T^{∗} is an extension of T_{∈} that ^{×}-interprets CZF. Then there is a functional G : o → o → o in T^{∗} with

(1) T^{∗} ⊢ R^{∈} mv(^{a}b) → ∃d^{∈}Gab R : a >−< d.

Proof: The ^{×}-interpretability of ∃c Full(c, a, b) gives terms C, Q[a, b] in T^{∗} with

T^{∗} ⊢ (∀q^{∈}Q≠∅) [R^{∈} mv(^{a}b) → ∃d^{∈}Cq R : a >−< d].

Define

Gab :≡ ⋃{Cq | q^{∈}Q}.

Then G fulfills (1). □ This is a corollary of 4.27 and lemma 2.2

Corollary 4.29 In T_{∈}^{+} an exponentiation functional exp with

T_{∈}^{+} ⊢ (f : a → b) → f^{∈} exp ab

is definable.

Proof: CZF proves the exponentiation theorem. The interpretation theorem and an argument similar to that in the theorem above gives the exponentiation functional. □

4.c Constructive set theories in all finite types

In this section we characterize the strength of the translation ^{×}. In order to obtain a framework for this discussion, we define systems of constructive set theory in all finite types, i.e., extensions of T_{∈} with quantifiers of arbitrary type. We introduce principles (WAC), (IP_{∈}) and (M_{∈}) that are strong enough to prove the equivalence φ ↔ φ^{×} for all formulae φ of the extended language. Vice versa these schemes are derivable using suitable instances of φ ↔ φ^{×}.

Definition 4.30 (Constructive set theories in all finite types) The language of T_{∈} extended by quantifiers for all finite types is called L_{∈}^{ω}. The extension with the fullness functional is called L_{∈}^{ω}^{+}. We define the following theories and schemes:

(i)

CZF^{ω−} :≡ T_{∈} [ but the (Ext-Rule) only with side-formulae φ ∈ T_{∈}] +

Q1 − Q4 for quantifiers of arbitrary finite type +

(Strong-Collection) and (Foundation) for arbitrary formulae of L_{∈}^{ω}.

(ii) CZF^{ω} is defined similar to CZF^{ω−}, but with T_{∈}^{+} instead of T_{∈} and L_{∈}^{ω}^{+} instead of L_{∈}^{ω}.

(iii) (WAC_{στ}) :≡ ∀x^{σ}∃y^{τ}φ(x, y) → ∃Y^{o→σ→τ}S^{σ→o}∀x(∀s^{∈}Sx≠∅) φ(x, Y sx)

(iv) (WAC) :≡ ⋃_{στ} (WAC_{στ})

(v) (IP_{∈}) :≡ (∀wφ → ∃uψ) → ∃US[∀wφ → (∀s^{∈}S≠∅) ψ(Us)] φ ∈ T_{∈} (vi) (M_{∈}) :≡ (∀wφ → ψ) → ∃W X[(∀x^{∈}X)φ(W x) → ψ] φ, ψ ∈ T_{∈}

(vii) CZF_{∗}^{ω−} :≡ CZF^{ω−} + (WAC) + (IP_{∈}) + (M_{∈})

(viii) CZF_{∗}^{ω} :≡ CZF^{ω} + (WAC) + (IP_{∈}) + (M_{∈}).

Note that in (iii), (v) and (vi) x^{σ}, y^{τ}, w and u may be tuples. In (viii) the conditions on the formulae within the schemes (WAC), (M_{∈}) and (IP_{∈}) are formulated, of course, with T_{∈}^{+} instead of T_{∈}.

Remark 4.31 (Equality in higher types and the (Ext-Rule)) In remark 4.19 we explained the role of equality in higher types and the (Ext-Rule) in T_{∈}. Of course, all these considerations apply here as well, in particular the need for the (Ext-Rule) to define e.g. ↾_{σ}. On the other hand, we obviously want our systems of constructive set theories in all finite types to be^{×}-interpretable. However, the (Ext-Rule) with arbitrary side-formulae is no longer ^{×}-interpretable (see the next section). Hence we accept the (Ext-Rule) only with side-formulae φ ∈ T_{∈}.

As we shall see in corollary 4.51, as a consequence of this limitation none of the theories CZF^{ω−}, CZF^{ω}, CZF_{∗}^{ω−} and CZF_{∗}^{ω} satisfy the deduction theorem.

Definition 4.32 (The translation ^{×} for L_{∈}^{ω}) The translation ^{×} can easily be extended to the language L_{∈}^{ω} (and similarly to L_{∈}^{ω}^{+}): to every formula φ ∈ L_{∈}^{ω} we assign a formula φ^{×} :≡ ∃vQ ∀w (∀q^{∈}Q^{≠∅}) φ̄(vq, w) with matrix (∀q^{∈}Q^{≠∅}) φ̄(vq, w) a formula of T_{∈}:

φ^{×} :≡ φ for φ in T_{∈}.

The rest of the definition is just as in definition 4.21.

The requirement that the matrix of a translated formula is in T_{∈} needs a comment. Formulae of T_{∈} contain restricted existential quantifiers only within Δ_{0}-formulae. In general φ̄ may contain equations of arbitrary type. Note that (∀q^{∈}Q^{≠∅}) φ̄(vq, w) is

(∃q^{∈}Q_{1})(q = q) ∧ … ∧ (∃q^{∈}Q_{n})(q = q) ∧ (∀q_{1}^{∈}Q_{1})..(∀q_{n}^{∈}Q_{n})φ̄(v_{1}q_{1}, .., v_{n}q_{n}, w_{1}, ..w_{m}),

and this is in T_{∈}, provided φ̄ is in T_{∈}.

Lemma 4.33 CZF^{ω−} derives (Pair), (Union), (Infinity) and (Δ_{0}-Separation). Moreover, CZF^{ω} derives (Subset-Collection).

Proof: (Pair), (Union), (Infinity) and (Δ_{0}-Separation) are all set-existence axioms. The sets all have a Δ_{0}-description, and T_{∈} contains terms for these sets. Hence CZF^{ω−} proves the existence of these sets. Since CZF^{ω} is augmented by the fullness functional we obtain the second claim. □

Lemma 4.34 In CZF^{ω−} we can prove:

(i) ∃v∀wφ(v, w) ↔ ∃V Q∀w(∀q^{∈}Q^{≠∅}) φ(V q, w).

(ii) ∃Y S∀x(∀s^{∈}Sx^{≠∅}) φ(x, Y sx) → ∀x∃yφ(x, y).

(iii) ∃US[∀wφ → (∀s^{∈}S≠∅) ψ(Us)] → (∀wφ → ∃uψ).

(iv) ∃W X[(∀x^{∈}X)φ(W x) → ψ] → (∀wφ → ψ).

Proof: (i). ′′ →′′: With Q :≡ 1 and V :≡ λq.v. ′′ ← “: Suppose there are V Q with ∀w(∀q^{∈}Q^{≠∅}) φ(V q, w). Then there is q_{0}^{∈}Q. Hence we have ∀wφ(V q_{0}, w) and so ∃v∀wφ(v, w). Note that v is in general not definable in terms of T_{∈}. (ii) is easy: Suppose that the antecedent holds and x is given. Then there is s^{∈}Sx. Hence y := Y sx fulfills φ(x, y). (iii) and (iv) are similar. □

We now give a characterization of the translation ^{×}, namely that CZF^{ω−} + {φ ↔ φ^{×} | φ ∈ L_{∈}^{ω}} = CZF_{∗}^{ω−}. One inclusion is proved by translating the additional principles of CZF_{∗}^{ω−} and using the equivalence of the principles to their translations. The other inclusion is proved by induction on the build-up of formulae: using the principles of CZF_{∗}^{ω−} we show the equivalence of any formula to its translation. The following technical lemma is useful in the induction step.

Lemma 4.35 CZF^{ω−} proves the equivalence of the following two formulae:

(1) ∃W XY RT S∀vQz(∀s^{∈}SvQ^{≠∅}) (∀t^{∈}T vQzs_{5}^{≠∅})
[(∀x^{∈}XvQzs_{2}t_{2})(∀q^{∈}Q^{≠∅}) φ̄(vq, W vQzs_{1}t_{1}x) → (∀r^{∈}RvQs_{4}^{≠∅}) ψ̄(Y vQs_{3}r, z)]

(2) ∃W XY R∀vQz[(∀x^{∈}XvQz)(∀q^{∈}Q^{≠∅}) φ̄(vq, W vQzx) → (∀r^{∈}RvQ^{≠∅}) ψ̄(Y vQr, z)].

Proof: (2) ⇒ (1) is immediate by setting SvQ :≡ 1, T vQzs_{5} :≡ 1. With a distribution argument 4.22 we obtain the other direction. □

Lemma 4.36 CZF^{ω−} proves for all formulae φ ∈ L_{∈}^{ω}:

(∀s^{∈}S≠∅) ∀wφ(w, s) ↔ ∀w(∀s^{∈}S≠∅) φ(w, s)

Proof: This is obvious. □ We are now able to prove the characterization theorem:

Theorem 4.37 (Characterization Theorem) CZF_{∗}^{ω−} = CZF^{ω−} + {φ ↔ φ^{×} | φ ∈ L_{∈}^{ω}}.

Proof: ′′ ⊆′′: We prove the additional schemes of CZF_{∗}^{ω−} in CZF^{ω−} + {φ ↔ φ^{×} | φ ∈ L_{∈}^{ω}}. (WAC): By φ ↔ φ^{×} we have φ ↔ ∃v∀wφ̃ for some φ̃ ∈ T_{∈}. Hence

(∀x∃yφ)^{×} ↔ (∀x∃yv∀wφ̃)^{×}
↔ ∃Y V ST ∀x∀w(∀s^{∈}Sx^{≠∅}) (∀t^{∈}T x^{≠∅}) φ̃(x, Y sx, V tx, w).

By lemma 4.34 (ii) this implies

∃Y S∀x(∀s^{∈}Sx^{≠∅}) ∃v∀wφ̃(x, Y sx, v, w) ↔ ∃Y S∀x(∀s^{∈}Sx^{≠∅}) φ(x, Y sx).

Using the equivalence of a formula to its translation this yields (WAC). (IP_{∈}): Suppose φ ∈ T_{∈} and ψ ↔ ∃y∀zψ̃ with ψ̃ ∈ T_{∈}.

(∀wφ → ∃uψ)^{×} ↔ (∀wφ → ∃uy∀zψ̃)^{×}
↔ ∃W XUY ST ∀z[(∀x^{∈}Xz)φ(W zx) → (∀s^{∈}S^{≠∅}) (∀t^{∈}T^{≠∅}) ψ̃(z, Us, Y t)]

By lemma 4.34 (ii), (iv) we have :

→ ∃UY ST ∀z[∀wφ → (∀s^{∈}S^{≠∅}) (∀t^{∈}T^{≠∅}) ψ̃(z, Us, Y t)]
→ ∃US[∀wφ → (∀s^{∈}S^{≠∅}) ∃y∀zψ̃(z, Us, y)].

This is now equivalent to

∃US[∀wφ → (∀s^{∈}S≠∅) ψ(Us)].

(M_{∈}): Suppose φ, ψ ∈ T_{∈}. Then

(∀wφ → ψ)^{×} ≡ ∃W X[(∀x^{∈}X)φ(W x) → ψ]

and we obtain immediately (M_{∈}).

′′ ⊇′′: We show for all φ ∈ L_{∈}^{ω} the equivalence φ ↔ φ^{×} by induction on φ.

1. φ ∈ T_{∈} is obvious.

2. φ ≡ φ_{0} ∧ φ_{1} and φ ≡ φ_{0} ∨ φ_{1}: immediately using the induction hypothesis.

3. φ ≡ ∃uφ_{0}: The assertion follows by induction hypothesis and lemma 4.34 (i).

4. φ ≡ ∀uφ_{0}(u). We have the following equivalences:

∀uφ_{0}(u) ↔_{I.H.} ∀u(φ_{0}(u)^{×}) ≡ ∀u∃vQ∀w(∀q^{∈}Q^{≠∅}) φ̄_{0}(vq, w, u).

Using (WAC) and 4.34 (ii) this is equivalent to

∃V QST ∀uw(∀s^{∈}Su≠∅) (∀t^{∈}T u≠∅) (∀q^{∈}Qtu≠∅) φ̄_{0}(V qsu, w, u).

By distribution this is equivalent to

∃V Q∀uw(∀q^{∈}Qu≠∅) φ̄_{0}(V qu, w, u)

and by 4.34 (i) we obtain (∀uφ_{0}(u))^{×}.

5. φ ≡ φ_{0} → ψ_{0}. We have:

φ ↔_{I.H.} φ_{0}^{×} → ψ_{0}^{×} ≡ ∃vQ∀w(∀q^{∈}Q^{≠∅}) φ̄_{0}(vq, w) → ∃yR∀z(∀r^{∈}R^{≠∅}) ψ̄_{0}(yr, z)

Intuitionistic predicate logic yields

∀vQ[∀w(∀q^{∈}Q^{≠∅}) φ̄_{0}(vq, w) → ∃yR∀z(∀r^{∈}R^{≠∅}) ψ̄_{0}(yr, z)].

By (IP_{∈}) and 4.34 (iii) this is equivalent to

∀vQ∃Y RST [∀w(∀q^{∈}Q^{≠∅}) φ̄_{0}(vq, w) → (∀s^{∈}S^{≠∅}) (∀t^{∈}T^{≠∅}) ∀z(∀r^{∈}Rs^{≠∅}) ψ̄_{0}(Y rt, z)].

Distribution and lemma 4.36 give

∀vQ∃Y R[∀w(∀q^{∈}Q^{≠∅}) φ̄_{0}(vq, w) → ∀z(∀r^{∈}R^{≠∅}) ψ̄_{0}(Y r, z)].

Intuitionistic predicate logic yields

∀vQ∃Y R∀z[∀w(∀q^{∈}Q^{≠∅}) φ̄_{0}(vq, w) → (∀r^{∈}R^{≠∅}) ψ̄_{0}(Y r, z)].

We apply (M_{∈}) and 4.34 (iv):

∀vQ∃Y R∀z∃W X[(∀x^{∈}X)(∀q^{∈}Q^{≠∅}) φ̄_{0}(vq, W x) → (∀r^{∈}R^{≠∅}) ψ̄_{0}(Y r, z)].

Two applications of (WAC) provide

∃W XY RT S∀vQz(∀s^{∈}SvQ^{≠∅}) (∀t^{∈}T vQzs_{5}^{≠∅})
[(∀x^{∈}XvQzs_{2}t_{2})(∀q^{∈}Q^{≠∅}) φ̄_{0}(vq, W vQzs_{1}t_{1}x) → (∀r^{∈}RvQs_{4}^{≠∅}) ψ̄_{0}(Y vQs_{3}r, z)]

By lemma 4.35 this is equivalent to φ^{×}. □

The next theorem shows that the principles (WAC), (IP_{∈}) and (M_{∈}) are interpretable in T_{∈}:

Theorem 4.38 (Interpretability of CZF_{∗}^{ω−}) CZF_{∗}^{ω−} is ^{×}-interpretable in T_{∈}.

Proof: The logical and set-theoretical axioms and rules of CZF^{ω−} are interpreted in the same way as in the case of CZF^{−}. For the introduction rule Q4 of the restricted existential quantifier note that in formulae of T_{∈} the restricted existential quantifier occurs only within Δ_{0}-formulae. Therefore the separation-terms needed for the interpretation of that rule are available. (Similarly, for A10 we use the fact that disjunctions occur only within Δ_{0}-formulae). All defining axioms of T_{∈} and the (Ext-Rule) are unchanged by the translation and hence interpretable. It remains to interpret the new principles of CZF_{∗}^{ω−}. By the characterization theorem 4.37 it suffices to show that for any φ ∈ L_{∈}^{ω} the formula φ ↔ φ^{×} is interpretable. Suppose φ^{×} ≡ ∃vQ ∀w (∀q^{∈}Q^{≠∅}) φ̄(vq, w) and further (φ^{×})^{×} ≡ ∃V QST ∀w(∀s^{∈}S^{≠∅}) (∀t^{∈}T^{≠∅}) (∀q^{∈}Qt^{≠∅}) φ̄(V qs, w).

1. φ → φ^{×}. We have to show that there are terms W_{0}, X, V_{1}, Q_{1}, S, T with

(∀x^{∈}Xv_{0}Q_{0}w_{1})(∀q^{∈}Q_{0}^{≠∅}) φ̄(v_{0}q, W_{0}v_{0}Q_{0}w_{1}x) →

(∀s^{∈}Sv_{0}Q_{0}^{≠∅}) (∀t^{∈}T v_{0}Q_{0}^{≠∅}) (∀q^{∈}Q_{1}v_{0}Q_{0}t^{≠∅}) φ̄(V_{1}v_{0}Q_{0}qs, w_{1}).

This follows by putting Xv_{0}Q_{0}w_{1} :≡ 1, Sv_{0}Q_{0} :≡ 1, T v_{0}Q_{0} :≡ 1, Q_{1}v_{0}Q_{0}t :≡ Q_{0}, W_{0}v_{0}Q_{0}w_{1}x :≡ w_{1}, V_{1}v_{0}Q_{0}qs :≡ v_{0}q.

2. φ^{×} → φ: We need terms W_{1}, X_{1}, V_{0}, Q_{0} with

(∀x^{∈}XV_{1}Q_{1}ST w_{0})(∀s^{∈}S^{≠∅}) (∀t^{∈}T^{≠∅}) (∀q^{∈}Q_{1}t^{≠∅}) φ̄(V_{1}qs, W_{1}V_{1}Q_{1}ST w_{0}x) →

4.c Constructive set theories in all finite types  75
(∀q^{∈}Q_{0}V_{1}Q_{1}ST^{≠∅}) φ̄(V_{0}V_{1}Q_{1}ST q, w_{0}).

The standard distribution argument provides the interpreting terms:

XV_{1}Q_{1}ST w_{0} :≡ 1, W_{1}V_{1}Q_{1}ST w_{0}x :≡ w_{0}, Q_{0}V_{1}Q_{1}ST :≡ {(q, s) | q^{∈}Q_{1}t, s^{∈}S, t^{∈}T }

V_{0}V_{1}Q_{1}ST q :≡ V_{1}(q)_{0}(q)_{1}.

This yields the theorem. □ Corollary 4.39 CZF_{∗}^{ω−} is a conservative extension of T_{∈} and it is consistent relative to T_{∈}. □ Corollary 4.40 Every formula that is ^{×}-interpretable in T_{∈} is derivable in CZF_{∗}^{ω−}. Proof: Suppose φ^{×} ≡ ∃vQ ∀w (∀q^{∈}Q^{≠∅}) φ̄(vq, w) and φ is interpretable in T_{∈}, i.e. we have terms v, Q with

(1) T_{∈} ⊢ (∀q^{∈}Q^{≠∅}) φ̄(vq, w). Then (1) is also derivable in CZF^{ω−}, hence

CZF^{ω−} ⊢ ∃vQ∀w(∀q^{∈}Q^{≠∅}) φ̄(vq, w). The corollary follows now from φ ↔ φ^{×}. □ Corollary 4.41 (Characterization of the set-functions and definable sets of CZF_{∗}^{ω−})

(i) Suppose CZF_{∗}^{ω−} ⊢ ∀x∃!y^{o}∃zφ(x, y, z) with φ ∈ T_{∈}. Then there are functionals F, S, Z

in T_{∈} with

T_{∈} ⊢ (∀s^{∈}Sx≠∅) φ(x, F x, Zxs).

(ii) Suppose CZF_{∗}^{ω−} ⊢ ∀x∃!y^{o}φ(x, y) with φ ∈ L_{∈}^{ω} arbitrary. Then there is a functional F

in T_{∈} with

CZF_{∗}^{ω−} ⊢ φ(x, F x).

(iii) Suppose CZF_{∗}^{ω−} ⊢ (∀x^{∈}a)∃!y^{o}∃zφ(x, y, z) with φ ∈ T_{∈}. Then there are functionals

F, R, Z in T_{∈} with

T_{∈} ⊢ x^{∈}a → (∀r^{∈}Rx≠∅) φ(x, F x, Zrx)

(iv) If CZF_{∗}^{ω−} ⊢ ∃xψ(x) then there are functionals S, X in T_{∈} with

CZF_{∗}^{ω−} ⊢ (∀s^{∈}S≠∅) ψ(Xs)

Proof: (i) and (ii) are analogous to 4.25. (iv) follows immediately from the interpretation theorem. (iii): Consider the ^{×}-translation of x^{∈}a → ∃y∃zφ(x, y, z). This yields

CZF_{∗}^{ω−} ⊢ ∃Y ZSR[x^{∈}a → (∀s^{∈}S≠∅) (∀r^{∈}R≠∅) φ(x, Y s, Zr)].

By the interpretation theorem and the same argument as in 4.25 we obtain the functionals F, Z and R. □ It is straightforward to get the results of this section also for CZF_{∗}^{ω} and T_{∈}^{+}:

Corollary 4.42 CZF_{∗}^{ω} = CZF^{ω} + {φ ↔ φ^{×} | φ ∈ L_{∈}^{ω}^{+}}. Furthermore, 4.38,4.39,4.40 and 4.41 also hold when we replace CZF_{∗}^{ω−} by CZF_{∗}^{ω} and T_{∈} by T_{∈}^{+}. □

4.d Hereditarily majorizable functionals

The aim of this section is to prove that there is no ^{×}-interpretation by functionals of T_{∈}^{+} of the simplest non-trivial case of the axiom of extensionality

(E2) ∀Y αβ(∀u[αu = βu] → Y α = Y β)

where α and β have type 1 and Y has type 2. The proof is similar to that given in [Ho73] and uses the notion of hereditarily majorizable functionals. Howard introduced this in the context of arithmetic. We adapt his definition to set theory.

Definition 4.43 (Hereditarily majorizable set functionals) We define a transitive ordering < on the universe of sets.

a < b :≡ a^{∈} TC(b).

By induction on σ we now define X^{∗} maj_{σ} X for functionals X^{∗}, X of type σ:

(i) a^{∗} maj_{o} a :≡ a^{∗} ⩾ a (:≡ a^{∗} > a ∨ a^{∗} = a)

(ii) F^{∗} maj_{σ→τ} F :≡ ∀GG^{∗}(G^{∗} maj_{σ} G ⇒ F^{∗}G^{∗} maj_{τ} F G)

We write F^{∗} maj F if F^{∗} and F are of the same type τ and F^{∗} maj_{τ} F. A functional is called hereditarily majorizable if there is a functional F^{∗} with F^{∗} maj F.

Lemma 4.44 The relation maj has the following properties

(i) F^{∗} maj F and G^{∗} maj G ⇒ F^{∗}G^{∗} maj F G

(ii) G_{r}^{∗} maj G_{r} for 0 ⩽ r ⩽ p implies G_{0}^{∗}G_{1}^{∗} … G_{p}^{∗} maj G_{0}G_{1} … G_{p}

(iii) Suppose HX_{1} … X_{n} is of type o. If H^{∗}X_{1}^{∗} … X_{n}^{∗} maj HX_{1} … X_{n} for all X_{i}^{∗}, X_{i} with X_{i}^{∗} maj X_{i}(i = 1, … n), then H^{∗} maj H. □

We will now extend the system T_{∈}^{+} to T_{∈}^{∗} by adding a new functional R^{+}. We prove that all functionals of this extension are hereditarily majorizable. Later we show that hereditarily majorizable functionals do not interpret (E2). This implies that (E2) is not ^{×}-interpretable in T_{∈}^{+}.

Definition 4.45 (The system T_{∈}^{∗}) T_{∈}^{∗} is the extension of T_{∈}^{+} which is equipped by a new functional R^{+} with defining axiom

R^{+}Ga = G[(R^{+}G)↾^{+}a]a

where the modified restriction functional ↾^{+} is defined

↾_{o}^{+}fax :≡ {fz | z < a}, ↾_{σ→τ}^{+}fax :≡ λz^{σ}.[((λx^{o}.fxz)↾_{τ}^{+}a)x].

In this section we argue in CZF^{ω} + T_{∈}^{∗} + (REM). Note that Howard’s proof (cf. [Tr73], pp. 454-461) makes essential use of the excluded middle for type o. Note further that we need the exponentiation functional to majorize the functional U and R^{+} to majorize R.

Theorem 4.46 (CZF^{ω} + T_{∈}^{∗} + (REM)) All functionals of T_{∈}^{+} are hereditarily majorizable by functionals of T_{∈}^{∗}.

Proof: (i): 0, ω, K and S are self-majorizing. (ii): The functional Suc is majorized by

Suc^{∗} :≡ λab.{Suc xy | x ⩽ a, y ⩽ b}.

Suppose we are given a^{∗}, a, b^{∗}, b with a^{∗} maj a, b^{∗} maj b. We have to show Suc ab ⩽ Suc^{∗} a^{∗}b^{∗}. By the premise a ⩽ a^{∗}, b ⩽ b^{∗} we have Suc ab^{∈} Suc^{∗} a^{∗}b^{∗}, hence Suc ab ⩽ Suc^{∗} a^{∗}b^{∗}. (iii): Similar we obtain majorizing functionals for N, I and the fullness-functional F:

N^{∗} :≡ λabc.{Nxyz | x ⩽ a, y ⩽ b, z ⩽ c}

I^{∗} :≡ λab.{Ixy | x ⩽ a, y ⩽ b}
F^{∗} :≡ λab.{F xy | x ⩽ a, y ⩽ b}.

(iv): With U the situation is different. We need a functional U^{∗} with the property that for f^{∗} maj f and a^{∗} maj a we have U^{∗}f^{∗}a^{∗} maj Ufa. That is ⋃{fz | z^{∈}a} ⩽ U^{∗}f^{∗}a^{∗}. However, it is not so easy to recover f from f^{∗} if we only know f^{∗} maj f. Therefore we need the following construction using the exponentiation-functional (which is definable in T_{∈}^{+} by 4.29):

U^{∗} :≡ λfa.{⋃ rng(g) | g^{∈}(exp x(TC(fa) ∪ {fa})), x ⩽ a}.

Suppose f^{∗} maj f and a^{∗} maj a. We are done if we prove Ufa = ⋃{fz | z^{∈}a}^{∈}U^{∗}f^{∗}a^{∗}. Let g := {< x, fx > | x^{∈}a}. For all x^{∈}a we have by assumption x ⩽ a^{∗} and hence with f^{∗} maj f it follows fx ⩽ f^{∗}a^{∗}. This implies that g is a function (i.e., a set and not a functional) with

g : a → TC(f^{∗}a^{∗}) ∪ {f^{∗}a^{∗}}

and so with ⋃ rng(g) = Ufa and ⋃ rng(g)^{∈}U^{∗}f^{∗}a^{∗} the majorizability is proved.

(v) For R suppose that G, X_{2}, …, X_{n} are of appropriate type such that RGaX_{2} … X_{n} is of type o. Then we define

R^{∗} :≡ λGaX_{2} … X_{n}.{R^{+}GxX_{2} … X_{n} | x ⩽ a},

and show that R^{∗} maj R. First we prove by ∈-induction on a that for all G^{∗} maj G

(1) R^{+}G^{∗}a maj RGa

holds. Suppose that by induction hypothesis (1) holds for all x^{∈}a. By the recursion equations for R^{+} and R we need to show

(2) G^{∗}(R^{+}G^{∗}↾^{+}a)a maj G(RG↾a)a.

By assumption G^{∗} maj G, therefore we have to show

(3) (R^{+}G^{∗}↾^{+}a) maj (RG↾a).

Using lemma 4.44 (i) we take X_{1}^{∗} maj X_{1}, …, X_{n}^{∗} maj X_{n} and show

(4) R^{+}G^{∗}↾^{+}aX_{1}^{∗} … X_{n}^{∗} maj RG↾aX_{1} … X_{n}.

By definition we have

R^{+}G^{∗}↾^{+}aX_{1}^{∗} … X_{n}^{∗} = {R^{+}G^{∗}xX_{2}^{∗} … X_{n}^{∗} | x < a}

and

RG↾aX_{1} … X_{n} = \{ \begin{matrix}RGX_{1} … X_{n}, & X_{1}^{∈}a \\ 0, & ¬X_{1}^{∈}a. \end{matrix}

Consider the two cases (with (REM) this case-distinction is complete): 1.: X_{1} ∉ a. Then obviously 0 ⩽ R^{+}G^{∗}↾^{+}aX_{1}^{∗} … X_{n}^{∗}. 2.: X_{1}^{∈}a. Then by induction hypothesis we have R^{+}G^{∗}X_{1} maj RGX_{1} and hence R^{+}G^{∗}X_{1}X_{2}^{∗} … X_{n}^{∗} maj RGX_{1} … X_{n}. Therefore RG↾aX_{1} … X_{n} ⩽ R^{+}G^{∗}↾^{+}aX_{1}^{∗} … X_{n}^{∗}. In both cases we have (4) and obtain (2), hence (1). Now we show that R^{∗} maj R: Let G^{∗} maj G, a^{∗} maj a, X_{2}^{∗} maj X_{2}, …, X_{n}^{∗} maj X_{n}. Then R^{∗}G^{∗}a^{∗}X_{2}^{∗} … X_{n}^{∗} = {R^{+}G^{∗}xX_{2}^{∗} … X_{n}^{∗} | x ⩽ a^{∗}}. With (1) we have RGaX_{2} … X_{n} ⩽ R^{+}G^{∗}aX_{2}^{∗} … X_{n}^{∗} and by definition this is ⩽ R^{∗}G^{∗}a^{∗}X_{2}^{∗} … X_{n}^{∗}. This concludes the proof of the theorem. □

In the following theorem we assume that X_{r} and Z_{1}, …, Z_{s(r)} are variables such that X_{r}Z_{1} … Z_{s(r)} is of type o.

Theorem 4.47 (CZF^{ω}) Let F : σ_{1} → … → σ_{p} → o be hereditarily majorizable, a : o and for 1 ⩽ r ⩽ p denote by M_{r} the set of all functionals X_{r} of type σ_{r} with

∀Z_{1} … Z_{s(r)} (X_{r}Z_{1} … Z_{s(r)} ⩽ a).

Then there is b : o with

(∀X_{1}^{∈}M_{1}) … (∀X_{p}^{∈}M_{p}) F X_{1} … X_{p} < b.

Proof: Take F^{∗} with F^{∗} maj F. For 1 ⩽ r ⩽ p let G_{r}^{∗} :≡ λZ_{1} … Z_{s(r)}.a. Then lemma 4.44 yields (∀x^{∈}M_{r}) G_{r}^{∗} maj x and hence:

(∀X_{1}^{∈}M_{1}) … (∀X_{p}^{∈}M_{p}) F^{∗}G_{1}^{∗} … G_{p}^{∗} maj F X_{1} … X_{p}.

The definition b :≡ {F^{∗}G_{1}^{∗} … G_{p}^{∗}} gives a sufficiently large majorant. □

Theorem 4.48 (CZF^{ω} + (REM)) For r = 1, 2 let N_{r} be the set of all functionals X : r with

(1) ∀Z (XZ ⩽ 1).

Suppose further that there are X : o, a : o and F : 2 → 1 → 1 → o → o with

(2) (∀Y^{∈}N_{2})(∀α^{∈}N_{1})(∀x^{∈}X) F Y α(λu.0)x < a.

Then the following does not hold:

(3) ∀Y αβ(Y α ≠ Y β → (∃x^{∈}X) α(F Y αβx) ≠ β(F Y αβx)).

Proof: Define the following functionals:

Y_{a} :≡ λα. \{ \begin{matrix}1, & if αa = 1 \\ 0, & else \end{matrix} and α_{b} :≡ λa. \{ \begin{matrix}0, & if a < b \\ 1, & else \end{matrix} .

Suppose that (3) holds for β :≡ λu.0 and let G :≡ λY αx.F Y α(λu.0)x. Then Y_{a}(λu.0) = 0 and Y_{a}(α_{a}) = 1, hence with (3) and β(GY_{a}α_{a}x) = 0 we obtain

(∃x^{∈}X) (α_{a}(GY_{a}α_{a}x) ≠ 0).

Therefore we have an x_{0}^{∈}X with ¬(F Y_{a}α_{a}(λu.0)x_{0} < a). This contradicts (2), since Y_{a}^{∈}N_{2}, α_{a}^{∈}N_{1} and x_{0}^{∈}X. □

Corollary 4.49 (CZF^{ω} + (REM)) There is no hereditarily majorizable functional that ^{×}-interprets the axiom of extensionality for type 2:

(E2) ∀u(αu = βu) → Y α = Y β.

Proof: Via distribution we have (E2)^{×} ↔ ∃F X[(∀x^{∈}X)(α(F Y αβx) = β(F Y αβx)) → Y α = Y β]. If (E2) is ^{×}-interpretable, then we have functionals F, X with

Y α ≠ Y β → (∃x^{∈}X)α(F Y αβx) ≠ β(F Y αβx).

If F is hereditarily majorizable, then by theorem 4.47 there is a with

(∀Y^{∈}N_{2})(∀α^{∈}N_{1})(∀x^{∈}X) F Y α(λu.0)x < a,

but this contradicts 4.48. □

Corollary 4.50 E2 is neither ^{×}-interpretable in T_{∈} nor in T_{∈}^{+}. Proof: Suppose it was. Then the interpreting functionals are hereditarily majorizable and this contradicts the previous corollary. □

Corollary 4.51 All theories CZF^{ω−}, CZF^{ω}, CZF_{∗}^{ω−}, CZF_{∗}^{ω} do not satisfy the deduction-theorem. Proof: Let S be any theory of CZF^{ω−}, CZF^{ω}, CZF_{∗}^{ω−}, CZF_{∗}^{ω} and α, β variables of type 1. Then

S + ∀u(αu = βu) ⊢ αu = βu, hence, with the (Ext-Rule)

S + ∀u(αu = βu) ⊢ α = β. For a contradiction, we now suppose that S satisfies the deduction theorem:

S ⊢ ∀u(αu = βu) → α = β. This gives (E2), which is not ^{×}-interpretable. On the other hand, S is ^{×}-interpretable (either in T_{∈} or in T_{∈}^{+}). □

Functional interpretation of Kripke-Platek set theories

In this chapter we define Kripke-Platek set theory in all finite types (KP ω^{ω}), introduce the Shoenfield variant ^{∨} of the translation ^{×} and show that KPω^{ω} is ^{∨}-interpretable in a classical version T_{∈}^{c} of T_{∈}. We further characterize the translation ^{∨} – here the crucial axiom is a schema related to (WAC). Similar to section 5 we are able to interpret KPω^{ω}_{∗} in T_{∈}^{c} and obtain a characterization of the provably total Σ-definable set functions of KPω^{ω}_{∗} in terms of functionals of T_{∈}^{c}. Finally we consider KPi, KPM and KPω + (Pow). We define additional functionals that are sufficient to interpret these theories. In this chapter we give the main definitions and sketch the results, usually the proofs are similar to those in the previous sections.

5.a Kripke-Platek set theories in all finite types

Definition 5.1 (The classical version T_{∈}^{c} of T_{∈}, the formula class Δ_{0}^{ω}) T_{∈}^{c} is the classical counterpart of T_{∈} and it is formulated in the {¬, ∨, (∀x^{∈}a)}-fragment of the language of T_{∈}. Terms of T_{∈}^{c} are those of T_{∈}. The formulae of T_{∈}^{c} are defined inductively:

(i) a^{∈}b, s =^{σ} t are formulae, where a, b are terms of type o and s, t terms of type σ.

(ii) If φ, ψ are formulae, then also (¬φ), (φ ∨ ψ) and (∀x^{∈}a)φ. We abbreviate as usual (∃x^{∈}a)φ :≡ ¬(∀x^{∈}a)¬φ; φ ∧ ψ :≡ ¬(¬φ ∨ ¬ψ); φ → ψ :≡ ¬φ ∨ ψ. Note that now disjunctions and restricted existential quantifiers may occur in front of equations of higher type. We denote this enlarged formula-class by Δ_{0}^{ω}. T_{∈}^{c} is T_{∈} plus the law of the excluded middle φ ∨ ¬φ for all formulae φ of T_{∈}^{c}. Similarly, we introduce the classical counterpart T_{∈}^{+c} of T_{∈}^{+}.

Note that we do not increase the number of basic functionals. In [Bu97] and [BH98] we introduced case distinction functionals as basic functionals, by lemma 4.10 we see that they are definable and hence superfluous. Note further that we have separation terms for all Δ_{0}-formulae, but not for formulae in Δ_{0}^{ω}.

Definition 5.2 (Kripke-Platek set theory in all finite types) The language of KP ω^{ω} is L_{∈}^{ω} (again we consider the {¬, ∨, ∀}-fragment). Axioms and rules are

- all axioms and rules of T_{∈}^{c}

- full classical predicate logic

- (Foundation) for all formulae of L_{∈}^{ω}

- (Δ_{0}^{ω}-Collection). Finally we define

KPω^{ω}_{∗} :≡ KPω^{ω} +{∀x^{σ}∃y^{τ}φ(x, y) → ∃Y^{o→σ→τ}S^{σ→o}∀x^{σ}(∃s^{∈}Sx)φ(x, Y sx) | φ ∈ Δ_{0}^{ω}}.

Note that this choice principle is slightly weaker than (WAC) for Δ_{0}^{ω}-formulae. This is, since we have only (∃s^{∈}Sx) instead of (∀s^{∈}Sx^{≠∅}) , but no (Δ_{0}^{ω}-Separation).

Lemma 5.3 KPω^{ω} ⊢ (Δ_{0}-Separation), (Pair), (Union), (Infinity). □

Hence KPω is a sub-theory of KPω^{ω}. The following translation ^{∨} is the Shoenfield variant of the translation ^{×}:

Definition 5.4 (The Shoenfield variant ^{∨} of ^{×}) To any formula of KPω^{ω} we assign a formula of the following shape: (where φ̄ ∈ Δ_{0}^{ω})

∀v_{1} … ∀v_{n}∃w_{1} … ∃w_{m}∃X_{1} … ∃X_{m}(∃x_{1}^{∈}X_{1}) … (∃x_{m}^{∈}X_{m})φ̄(v_{1}, …, v_{n}, w_{1}x_{1}, …, w_{m}x_{m}).

We abbreviate a formula of that shape by

∀v∃wX(∃x^{∈}X) φ̄(v, wx)

and define inductively:

φ^{∨} :≡ φ, φ ∈ Δ_{0}^{ω}
Now let φ^{∨} ≡ ∀v∃wX(∃x^{∈}X) φ̄(v, wx); ψ^{∨} ≡ ∀y∃zU(∃u^{∈}U) ψ̄(y, zu)

with disjoint tuples of variables

(φ ∨ ψ)^{∨} :≡ ∀vy∃wzXU(∃x^{∈}X)(∃u^{∈}U)[φ̄(v, wx) ∨ ψ̄(y, zu)]
(∀uφ)^{∨}  :≡ ∀vu∃wX(∃x^{∈}X) φ̄(u, v, wx)
(¬φ)^{∨}  :≡ ∀W X∃V Z(∃z^{∈}Z)[(∀x^{∈}X(V z)) ¬φ̄(V z, (W (V z))x)];

An L_{∈}^{ω}-formula φ with ^{∨}-translation

φ^{∨} ≡ ∀v∃wX(∃x^{∈}X) φ̄(v, wx)

is called ^{∨}-interpretable in T_{∈}^{c}, if there exist sequences of terms w, X with parameters v and the free variables of φ such that

T_{∈}^{c} ⊢ (∃x^{∈}X) φ̄(v, wx)

Remark: ^{∨} is a so-called ∀∃-translation and in contrast to the ∃∀-translation ^{∨}-interpreting terms for a formula φ may contain the parameters v.

Theorem 5.5 (Characterization of ^{∨}) KPω^{ω}_{∗} = KPω^{ω} +{φ ↔ φ^{∨} | φ ∈ L_{∈}^{ω}}.

Proof: The direction from the left to the right is easy by induction on the build-up of φ. The only interesting case is negation. ∀x^{σ}∃y^{τ}φ(x, y) → ∃Y^{o→σ→τ}S^{σ→o}∀x^{σ}(∃s^{∈}Sx)φ(x, Y sx) is precisely the axiom we need here. For the other direction consider the translation of the choice principle ∀x^{σ}∃y^{τ}φ(x, y) → ∃Y^{o→σ→τ}S^{σ→o}∀x^{σ}(∃s^{∈}Sx)φ(x, Y sx). □ This lemma is used in the proof of the interpretation theorem:

Lemma 5.6 Let φ^{∨} ≡ ∀v∃wX(∃x^{∈}X) φ̄(v, wx); ψ^{∨} ≡ ∀y∃zU(∃u^{∈}U)ψ̄(y, zu). Then these two claims are equivalent:

(i) φ ∨ ψ is ^{∨}-interpretable in T_{∈}^{c}.

(ii) There are terms w, X, z, U in T_{∈}^{c} such that

T_{∈}^{c} ⊢ (∃x^{∈}X) φ̄(v, wx) ∨ (∃u^{∈}U)ψ̄(y, zu)

Proof: Only (ii) ⇒ (i) needs a comment: Suppose (ii) holds and either X or U is the empty set. Then we define X′ :≡ X ∪ {0}, U′ :≡ U ∪ {0}. □

Theorem 5.7 (Interpretation Theorem) KPω^{ω}_{∗} is ^{∨}-interpretable in T_{∈}^{c}.

Proof: We proceed by induction on derivations in KPω^{ω}_{∗}: Logical axiom ¬φ ∨ φ: We have to find sequences of terms V, Z, w, X[W, X̃, v] (this means: V, Z, w and X may all contain the parameters W, X̃ and v) such that the following holds:

(∃x^{∈}X)φ̄(v, wx) ∨ (∃z^{∈}Z)(∀x^{∈}X̃(V z)) ¬φ̄(V z, (W (V z))x).

We choose Z :≡ 1; X :≡ X̃v; V z :≡ v; wx :≡ W vx.

(Substitution): ¬∀xφ(x) ∨ φ(a). We have to find sequences of terms w, X, V, Z, T, S[W, Z̃, v] such that the following holds:

(∃s^{∈}S)(∃t^{∈}T)(∀z^{∈}Z̃(V t)(Zs)) ¬φ̄(Zs, V t, (W (V t)(Zs))z) ∨ (∃x^{∈}X) φ̄(a, v, wx).

We choose: Zs :≡ a; V t :≡ v; S :≡ 1; T :≡ 1; X :≡ Z̃va and wx :≡ W vax.

(Associativity) and (Expansion) are obvious.

(Contraction): φ ∨ φ ⊢ φ: The induction hypothesis provides sequences of terms w, w′, X, X′[v, v′] such that

(∃x^{∈}X)φ̄(v, wx) ∨ (∃x^{∈}X′)φ̄(v′, w′x)

holds. We define

X̃ :≡ {(0, x) | x^{∈}X} ∪ {(1, x) | x^{∈}X′}

and

w̃x :≡ \{ \begin{matrix}w(x)_{1} & if (x)_{0} = 0 \\ w′(x)_{1} & else. \end{matrix}

The induction hypothesis yields for v :≡ v, v′ :≡ v:

(∃x^{∈}X̃) φ̄(v, w̃x).

(∀-Introduction). The interesting case is that the main formula is in Δ_{0}: (a^{∈}b → φ(a)) ∨ ψ ⊢ (∀x^{∈}b)φ(x) ∨ ψ, provided a is not free in ψ and φ ∈ Δ_{0}. By induction hypothesis exist sequences of terms z, U[y, a, b] such that

(IH) a^{∈}b → φ(a) ∨ (∃u^{∈}U)ψ̄(y, zu, b).

We define Ũ[y, b] :≡ {(u, a) | u^{∈}U, a^{∈}b} and z̃[y, b]x :≡ z[y, (x)_{1}, b](x)_{0}. This yields:

(∀x^{∈}b)φ(x) ∨ (∃u^{∈}Ũ)ψ̄(y, z̃u, b).

(Cut): Suppose that (ψ ∨ χ) is inferred from φ ∨ ψ and ¬φ ∨ χ. By induction hypothesis we have terms w, X, z, U[v, y] such that

(1) (∃x^{∈}X)(∃u^{∈}U) (φ̄(v, wx) ∨ ψ̄(y, zu))

and further terms V̂ , T, r, S[Ŵ , X̂, q] such that

(2) (∃t^{∈}T)(∃s^{∈}S) [(∀x^{∈}X̂(V̂ t))¬φ̄(V̂ t, (Ŵ (V̂ t))x) ∨ χ̄(q, rs))]

holds. We have to find terms z̃, Ũ, r̃, S̃[y, u] such that

(3) (∃u^{∈}Ũ)(∃s^{∈}S̃) [ψ̄(y, z̃u) ∨ χ̄(q, r̃s)].

We choose a sequence V of new variables of appropriate type. By putting v :≡ V t (1) is equivalent to (we denote this substitution by ′)

(4) (∀t^{∈}T)(∃x^{∈}X′)φ̄(V t, w′x) ∨ (∃t^{∈}T)(∃u^{∈}U′)ψ̄(y, z′u).

We define U′′ :≡ {(u, t) | u^{∈}U′[V t, y], t^{∈}T } and zx :≡ z[V (x)_{1}, y](x)_{0} and obtain:

(5) (∀t^{∈}T)(∃x^{∈}X′) φ̄(V t, w′x) ∨ (∃u^{∈}U′′)ψ̄(y, z′u).

We define fyv :≡ w′. Substituting λξ.X[ξ, y] for X̂ and fy for Ŵ in (5) and (2) yields

(6) (∀t^{∈}T′)(∃x^{∈}X′′) φ̄(V t, fy(V t)x) ∨ (∃u^{∈}Û)ψ̄(y, ẑu),
(7) (∃t^{∈}T′)(∀x^{∈}X[V̂′t, y])¬φ̄(V̂′t, fy(V̂′t)x) ∨ (∃s^{∈}Ŝ) χ̄(q, r̂s))

Finally, we set V̂′ for V in (6) and get

(8) (∀t^{∈}T′)(∃x^{∈}X[V̂′t, y])φ̄(V̂′t, fy(V̂′t)x) ∨ (∃u^{∈}Û)ψ̄(y, ẑu)).

From (7) and (8) now follows (3). All axioms of T_{∈}^{c} are not changed by the translation. Foundation is similar to section 4.b. (Δ_{0}^{ω}-Collection): Since we have classical logic it suffices to interpret the following rule:

(∀x^{∈}a)∃wφ(x, w) ∨ ψ ⊢ ∃v(∀x^{∈}a)(∃w^{∈}v)φ(x, w) ∨ ψ,

where φ ∈ Δ_{0}^{ω}. The induction hypothesis provides terms W, T [x, y, a] and sequences of terms z, U[x, y, a] such that

(IH) (∃t^{∈}T)(x^{∈}a → φ(x, W t)) ∨ (∃u^{∈}U)ψ̄(y, zu).

We have to find S̃, T̃, z̃, Ũ[y, a] such that

(1) (∃r^{∈}T̃)((∀x^{∈}a)(∃w^{∈}S̃r) φ(x, w)) ∨ (∃u^{∈}Ũ)ψ̄(y, z̃u).

We define

T̃ :≡ 1, S̃r :≡ {W t | t^{∈}T, x^{∈}a}, Ũ :≡ {(u, x) | u^{∈}Ũ, x^{∈}a}

and

z̃u :≡ z[(u)_{1}, y](u)_{0}.

We show (1): Suppose that ∀u^{∈}Ũ¬ψ̄(y, z̃u) holds. Let x^{∈}a. Using (IH) we find t^{∈}T such that φ(x, W t). Hence W t^{∈}S̃0 and we get

(∀x^{∈}a)(∃w^{∈}S̃0) φ(x, w).

Finally, we consider the schema ∀x^{σ}∃y^{τ}φ(x, y) → ∃Y^{o→σ→τ}S^{σ→o}∀x^{σ}(∃s^{∈}Sx)φ(x, Y sx). Due to the characterization theorem 5.5 it suffices to interpret φ ↔ φ^{∨}. This is straightforward by a distribution argument. □

Corollary 5.8 KPω^{ω}_{∗} is a conservative extension of T_{∈}^{c}, it is consistent relative to T_{∈}^{c}. If φ is ^{∨}-interpretable in T_{∈}^{c}, then φ is derivable in KPω^{ω}_{∗}.

Proof: Similar to chapter 4. □

Corollary 5.9 Suppose KPω^{ω}_{∗} ⊢ ∀x∃!y^{o}∃zφ(x, y, z) with φ ∈ Δ_{0}. Then there are functionals F, S, Z in T_{∈}^{c} such that

T_{∈}^{c} ⊢ (∃s^{∈}Sx)φ(x, F x, Zxs).

Proof: Similar to corollary 4.41. □

Corollary 5.10 The provably total Σ-definable set functions of KPω^{ω}_{∗} coincide with the type 1 functionals of T_{∈}^{c}.

Proof: Combine 5.9 with theorem 4 of [Bu97]. □

5.b KPi, KPM, KPω + (Pow)  89

5.b KPi, KPM, KPω + (Pow)

In the remaining part of this chapter we examine extensions of KPω and give interpretations using additional functionals. The results are straightforward: one just introduces new functionals that interpret the new axioms. Therefore these results are of minor interest, but for sake of completeness we briefly sketch them. We first consider the following two extensions of KPω: KPi and KPM. Recall the definition from 2.b.2. We introduce two new functionals + of type o → o and M of type 1 → o with the following defining axioms (write x^{+} for ^{+}x):

(+) Ad(x^{+}) ∧ x^{∈}x^{+}

(M1) Ad(Mf)

(M2) (∀x^{∈}Mf)(∃y^{∈}Mf)(fx = y)

Theorem 5.11 (i) KPi is ^{∨}-interpretable in T_{∈}^{c} + (Ad1) − (Ad3) plus the axiom (+). (ii) KPM is ^{∨}-interpretable in T_{∈}^{c} + (Ad1) − (Ad3) + (M1) + (M2).

Proof: (i) The translation does not change (Ad1) − (Ad3). The axiom (Lim) is obviously interpretable using the functional ^{+}. (ii) Using the Mahlo-functional we can interpret the Mahlo-rule: Suppose ψ^{∨} ≡ ∀v∃wX(∃x^{∈}X) ψ̄(v, wx). If the premise of (Mahlo) is interpretable we have terms w_{0}, X_{0}, Y_{0}, Q_{0}[a, v] such that

(1) ⊢ ∃x^{∈}X_{0}ψ̄(v, w_{0}x) ∨ (∃q^{∈}Q_{0})φ(a, Y_{0}q).

We need terms w, X, Z, Q[v] such that

(2) ⊢ ∃x^{∈}Xψ̄(v, wx) ∨ (∃q^{∈}Q)[Ad(Zq) ∧ (∀a^{∈}Zq)(∃y^{∈}Zq)φ(a, y)].

We define Y :≡ {Y_{0}q | q^{∈}Q_{0}} (note: Y contains the free variable a), Q :≡ 1, Zq :≡ M(λa.Y), X :≡ {(x, r) | x^{∈}X_{0}[a := r], r^{∈}Z0} and W x :≡ W_{0}[a := (x)_{1}](x)_{0}. We show (2): Suppose that we have

(3) (∀x^{∈}X)¬ψ̄(v, wx).

By (M1) we have Ad(Z0). Given a^{∈}Z0 we have to find y^{∈}Z0 with φ(a, y). By (3) we have (∀x^{∈}X_{0})¬ψ̄(v, w_{0}x), hence by (1) there is q^{∈}Q_{0} with φ(a, Y_{0}q). Then using (M2) we obtain y := Y_{0}q^{∈}Y ^{∈}M(λa.Y) = Z0. By Ad(Z0) the set Z0 is transitive and therefore y^{∈}Z0. This yields the assertion. □ Finally we consider the power-set axiom:

(Pow) ∃x∀y(y^{∈}x ↔ y ⊆ a). We obtain

Theorem 5.12 KPω + (Pow) is ^{∨}-interpretable in T_{∈}^{+c}. Proof: Recall that CZF proves the exponentiation axiom. In the presence of classical logic this is equivalent to (Pow). □

Part II Arithmetic On relations between classical and intuitionistic arithmetic

The subject of the second part of this thesis is intuitionistic first order arithmetic (Heyting arithmetic HA)^{1} and relations to its classical counterpart (Peano arithmetic PA)^{2} with a special focus on fragments of HA. HA is formulated in the first-order language L_{A} of arithmetic with non-logical symbols 0, suc, +, ·, <. Besides defining axioms, HA is equipped with the induction schema φ(0) ∧ ∀x(φ(x) → φ(x + 1)) → ∀xφ(x) for all formulae φ ∈ L_{A}. For Γ a class of formulae iΓ denotes Heyting arithmetic with induction restricted to formulae of Γ. Similarly, IΓ denotes the classical counterpart.

The canonical and well-investigated fragments of PA are IΠ_{n} (or IΣ_{n}, both coincide). The family (IΠ_{n})_{n∈ω} forms a proper hierarchy and the union of all IΠ_{n} is PA. This is due to the prenex normal-form theorem for classical predicate logic: every formula is equivalent to a formula in some Π_{n}. It is well known that the prenex normal-form theorem does not hold for intuitionistic logic, i.e., there are formulae that are not equivalent to any prenex formula. Thus the fragments iΠ_{n} do not a priori form a hierarchy that exhausts HA. Indeed, prenex induction within an intuitionistic framework is remarkably weak. Visser proved (we cite a sharpened version by Wehmeier [Weh96]): iΔ_{0}+ prenex induction (and hence each iΠ_{n+2}) is Π_{2}-conservative over iΠ_{2}. Since HA and PA are Π_{2}-equivalent, this implies that a measure for the complexity of a formula should not merely count quantifier alternations. Within intuitionistic arithmetic nested implication and quantification are responsible for the complexity of a formula. This leads to the subject of chapter 7 where we define classes Φ_{n} of formulae with the following properties:

1Standard sources are [Tr73] and [TvD88]

2A standard source is [Ka91]

(i) IΠ_{n} = IΦ_{n}.

(ii) ⋃_{n∈ω} Φ_{n} = L_{A} (modulo equivalence in iΔ_{0})

(iii) IΠ_{n} and iΦ_{n} prove the same Π_{2}-formulae.

Hence iΦ_{n} are reasonable counterparts to the classical fragments IΠ_{n}. Furthermore, the definition of Φ_{n} provides a normal-form for formulae of arithmetic in the intuitionistic context. In chapter 7 we also define a variant Φ′_{n} of our hierarchy such that all formulae of Φ′_{n} are almost negative. It turns out that iΦ_{n} and iΦ′_{n} are Π_{2}-equivalent. That means: To obtain the full Π_{2}-strength of HA one needs induction only for almost negative formulae. The definition of the formula-classes Φ′_{n} was inspired by the following considerations: The theories iΦ′_{n} have to prove the totality of all ≺ ω_{n+1}-descent-recursive functions (since these are precisely the IΠ_{n}-provably recursive functions). Inspection of the proof of the totality of these functions in IΠ_{n} yields that one needs induction for formulae obtained by iterated applications of the jump-operator to a Π_{2}-formula. (The iterated jump-operator is defined as follows φ(α)^{(0)} :≡ φ; φ(α)^{(n+1)} :≡ ∀γ [∀β ≺ γ φ(β)^{(n)} → (∀β ≺ γ + ω^{α}) φ(β)^{(n)}], where α, β, γ range over codes for ordinals less than ϵ_{0}; for more information see 7.38.) The classes Φ′_{n} are defined in a way that they contain those iterated jump-formulae.

The proof of the fact that IΠ_{n} is Π_{2}-conservative over iΦ_{n} needs some comments. Recall the well-known proof of the Π_{2}-conservativity of PA over HA (see 2.7): negative translation combined with Friedman-translation. However, in our case this does not apply. For given formula classes Γ_{1}, Γ_{2} the following condition is sufficient to make the Friedman-translation work. Provided

φ ∈ Γ_{1} and ψ ∈ Σ_{1} ⇒ (φ^{−})^{ψ} ∈ Γ_{2},

then IΓ_{1} is Π_{2}-conservative over iΓ_{2}. Note that PA and HA trivially fulfill these conditions, but even for Σ_{1} these properties fail. A refined translation was introduced by Coquand and Hofmann ([CH98], see also [Av98a]) and we obtain the Π_{2}-conservativity result by an adaptation of this technique. In section 7.6 we discuss variations of this proof.

We already mentioned the theorem by Visser and Wehmeier about prenex induction. In section 7.d we generalize that result and show that it holds at any level of our hierarchy: if we close Φ_{n} (for n ⩾ 2) under quantification (we call these new classes Θ_{n}) the new fragments iΘ_{n} still prove the same Π_{2}-theorems as iΦ_{n}. We obtain this result by a Dialectica interpretation due to Parsons of iΘ_{n} in appropriate fragments of Gödel’s T. As a corollary we obtain: iΘ_{n}, iΦ_{n} and IΠ_{n} all prove the same Π_{2}-sentences. Hence the provably recursive functions of these theories are precisely the ≺ ω_{n+1}-descent recursive functions.

When it comes to provably total functions (i.e., functions that are definable by arbitrary formulae; in contrast, provably recursive functions are Σ_{1}-definable) the situation is different. Classical predicate logic already proves the totality of a non-recursive function, namely the characteristic function of the Σ_{1}-complete set K. However, for intuitionistic theories it often happens that classes of provably recursive and of provably total functions coincide (this holds for example for HA and iΣ_{1}). The same is true for our fragments iΘ_{n+1} and iΦ_{n+1}: whenever ∀x∃yφ(x, y) is provable in one of them, we find a ≺ ω_{n+2}-descent recursive function f with ∀xφ(x, f(x)) (this is provable in HA^{ω∗}, see 2.19). We obtain this result by an easy application of Parsons interpretation theorem combined with a computability result for the fragments of T.

In chapter 8 we consider collection principles in the framework of intuitionistic arithmetic. The collection schema Bφ for φ is

Bφ :≡ ∀x ⩽ a∃yφ(x, y) → ∃z∀x ⩽ a∃y ⩽ zφ(x, y).

HA proves Bφ for arbitrary φ. The premise of the collection schema says that the (possibly multi-valued) function determined by φ is total on {0, …, a} and we already mentioned: in the intuitionistic framework it often happens that functions definable by arbitrary φ are not more complex than Σ_{1}-definable ones. This is already a hint that the strength of the collection schema does not depend on the formula complexity of φ - in contrast to classical arithmetic where we have IΔ_{0} + BΣ_{n+2} ⊢ IΣ_{n+1} and hence IΔ_{0} plus arbitrary collection is equal to PA. The main result of the first part of chapter 8 shows that collection within intuitionistic arithmetic is indeed rather weak: all provable total functions of iΔ_{0}+{Bφ | φ ∈ L_{A}} are bounded by polynomials. We prove this result by a functional interpretation in a weak fragment of T. This fragment is not equipped with recursors, but with some basic functionals and a maximum-functional Max. Max is already sufficient to interpret collection. The idea is the following: by the premise we are given a function f that is total on a. We interpret the conclusion by taking Max{fx | x < a}. A result by Kohlenbach now shows that all type 1 functionals of the interpreting system are bounded by polynomials.

In the final section we consider the contrapositive of the collection schema

Cφ :≡ ∀z∃x ⩽ a∀y ⩽ zφ(x, y) → ∃x ⩽ a∀yφ(x, y).

With classical logic Cφ follows from B¬φ, of course by using the law of excluded middle. It turns out that vice versa Cφ implies instances of the ψ ∨ ¬ψ. Consider for example Cφ with φ primitive recursive and a = 1:

(10) ∀z∃x ⩽ 1∀y ⩽ zφ(x, y) → ∃x ⩽ 1∀yφ(x, y).

Roughly speaking this schema turns infinitely many finite decisions (for every z the premise of (10) is decidable) into an infinite one. We use this idea to prove

iΔ_{0} + {Bφ, Cφ | φ ∈ L_{A}} ⊢ ψ ∨ ¬ψ

and hence iΔ_{0} + {Bφ, Cφ | φ ∈ L_{A}} = PA.

Fragments of Heyting arithmetic

Remark: The definitions of the classes Φ′_{n} and Φ_{n} of this chapter differ from those given in [Bu].

7.a Fragments iΦ′_{n}

In this section we define formula classes Φ′_{n} ⊆ L_{A} with the following properties: (i) IΠ_{n} = IΦ′_{n} (ii) IΠ_{n} and iΦ′_{n} prove the same Π_{2} formulae.

Definition 7.1 (Formula classes Φ′_{n}) (i) Φ′_{0} := Δ_{0} (ii) Φ′_{1} := Σ_{1} (iii) Suppose Φ′_{n} is already defined for n ⩾ 1. Then Φ′_{n+1} consists of all formulae

∀x(ψ → χ) where ψ ∈ Φ′_{n}, χ ∈ Σ_{1} and x is a finite tuple of variables which may occur both in ψ and χ.

Lemma 7.2 Every φ ∈ Φ′_{2} is equivalent (in iΔ_{0}) to a Π_{2}-formula. Proof: Suppose ψ_{0}, χ_{0} are in Δ_{0}. Then we have in iΔ_{0}:

∀x(∃yψ_{0} → ∃zχ_{0}) ↔ ∀xy∃z(ψ_{0} → χ_{0}) and this is equivalent to a Π_{2}-formula. □

Lemma 7.3 Suppose n ⩾ 2. Then we have within IΔ_{0} (i) Every φ ∈ Φ′_{n} is equivalent to a formula in Π_{n} (ii) Every φ ∈ Π_{n} is equivalent to a formula in Φ′_{n}. Proof: (i) By induction on n. The case n = 2 holds by the previous lemma. Suppose n ⩾ 2 and φ ≡ ∀x(ψ → χ) ∈ Φ′_{n+1} according to the definition of Φ′_{n+1}. By induction hypothesis ψ is equivalent to a formula in Π_{n}. We have within IΔ_{0}:

∀x(ψ → χ) ↔ \underbrace{∀x(\underbrace{¬ψ}_{Σ_{n}} ∨ \underbrace{χ}_{Σ_{1}})}_{Π_{n+1}}

(ii) We show by induction on k ⩾ 1 within classical logic:

- Every φ ∈ Π_{2k} is equivalent to a formula in Φ′_{2k}

- Every φ ∈ Π_{2k+1} is equivalent to a formula in Φ′_{2k+1} For k = 1 the first case holds by definition. For the second case note that classically we have for φ ∈ Δ_{0}

∀x∃y∀zφ ↔ ∀x¬∀y∃z¬φ ↔ \underbrace{∀x(\underbrace{∀y∃z¬φ}_{Φ′_{2}} → \underbrace{⊥}_{Σ_{1}})}_{Φ′_{3}}

For the induction step suppose that φ ∈ Π_{2k}. By induction hypothesis this is equivalent to a formula in Φ′_{2k} and we obtain

∀x∃yφ ↔ \underbrace{∀x(\underbrace{∀y(φ → ⊥)}_{Φ′_{2k+1}} → ⊥)}_{Φ′_{2(k+1)}}

The odd case is similar. □

Lemma 7.4 Suppose n ⩾ 2. Then we have in iΔ_{0}:

(i) If φ ∈ Φ′_{n}, then ∀xφ is equivalent to a formula in Φ′_{n}.

7.a Fragments iΦ′_{n}  99

(ii) If φ_{0}, φ_{1} ∈ Φ′_{n}, then φ_{0} ∧ φ_{1} is equivalent to a formula in Φ′_{n}

(iii) If φ_{0} ∈ Φ_{n−1}, φ_{1} ∈ Φ′_{n}, then φ_{0} → φ_{1} is equivalent to a formula in Φ′_{n}.

Proof: We proceed by induction on n.

1. n = 2. By lemma 7.2 all formulae of Φ′_{2} are equivalent to a formula in Π_{2}. Then (i) holds just by contracting two universal quantifiers. (ii) is straightforward. For (iii) we have within iΔ_{0} for φ, ψ ∈ Δ_{0}:

(∃zφ → ∀x∃yψ) ↔ ∀xz(φ → ∃yψ) ↔ ∀xz∃y(φ → ψ).

2. Now suppose n > 2. Again (i) is obvious. For (ii) take ψ_{0}, ψ_{1} ∈ Φ′_{n−1} and χ_{0}, χ_{1} ∈ Σ_{1}:

∀x(ψ_{0} → χ_{0}) ∧ ∀y(ψ_{1} → χ_{1}) ↔

∀zxy[(z = 0 → ψ_{0} ∧ z ≠ 0 → ψ_{1}) → (z = 0 → χ_{0} ∧ z ≠ 0 → χ_{1})].

This is in Φ′_{n} by induction hypothesis. To prove (iii) we take φ_{0} ∈ Φ′_{n−1} and φ_{1} ∈ Φ′_{n}. Then we have ψ_{1} ∈ Φ′_{n−1} and χ_{1} ∈ Σ_{1} with φ_{1} ≡ ∀y(ψ_{1} → χ_{1}). We obtain using the induction hypothesis

(φ_{0} → φ_{1}) ↔ [φ_{0} → ∀y(ψ_{1} → χ_{1})] ↔

[∀y(φ_{0} → (ψ_{1} → χ_{1}))] ↔ [\underbrace{∀y((\underbrace{φ_{0} ∧ ψ_{1}}_{Φ′_{n−1}}) → χ_{1})}_{Φ′_{n}}].

This concludes the proof of the lemma. □

Corollary 7.5 Every almost negative formula is equivalent to a formula in some Φ′_{n}. Hence the formulae in Φ′_{n} are normal forms for almost negative formulae. □

In the next subsection we establish the following theorem:

Theorem 7.6 IΠ_{n} is Π_{2}-conservative over iΦ′_{n}. □

Together with 7.3 this yields Corollary 7.7 iΦ′_{n} and IΠ_{n} prove the same Π_{2}-formulae. □

Corollary 7.8 HA and iΔ_{0} + {Iφ | φ almost negtive} are Π_{2}-equivalent. □

Corollary 7.8 follows also from the fact that HA is conservative over its negative fragment. By lemma 7.2 we have:

Corollary 7.9 IΠ_{2} is Π_{2}-conservative over iΠ_{2}. □

Corollary 7.10 iΦ′_{n} is closed under Markov’s rule for Σ_{1}-predicates, i.e.,

iΦ′_{n} ⊢ ¬¬∃xφ(x) ⇒ iΦ′_{n} ⊢ ∃xφ(x).

holds for φ ∈ Δ_{0}

Proof: Suppose iΦ′_{n} ⊢ ¬¬∃xφ(x). Then IΠ_{n} ⊢ ∃xφ(x). Corollary 7.7 gives iΦ′_{n} ⊢ ∃xφ(x).□

Corollary 7.9 shows that within the family of theories (iΠ_{n})_{n∈ω} there are two remarkable theories, namely iΠ_{0} and iΠ_{2}. They are the only ones that are Π_{2}-equivalent to their classical counterparts. For n = 1 and n ⩾ 3 this is not the case: the provably recursive functions of iΠ_{1} are bounded by polynomials (Wehmeier [Weh96]), and the union of all iΠ_{n} is Π_{2}- conservative over iΠ_{2} (Visser and Wehmeier [Weh96]).

The Proof of 7.6

The standard way to establish the Π_{2}-conservativity of a classical theory over its intuitionistic counterpart is via negative- and Friedman-translation (cf. 2.7). However, when it comes to fragments of arithmetic these methods do not always apply. Therefore we have to use an extended version of the Friedman-translation introduced by Coquand and Hofmann ([CH98], section 2.2). A variant of this technique and extensions of this method to Kripke-Platek set theory and bounded arithmetic were developed by Avigad, see [Av98a]. Before we start with the proof we should mention that for n ⩾ 1 the theory iΦ′_{n} contains iΣ_{1}. It is well-known that all primitive-recursive functions are definable and provably total in iΣ_{1}. Hence adding function-symbols for primitive-recursive functions together with their defining axioms extends iΦ′_{n} conservatively. Therefore we may freely use function-symbols for primitive-recursive functions in this subsection.

We fix a primitive-recursive enumeration of all (not necessarily closed) Σ_{1}-formulae, i.e., a mapping

∃zφ(z, x_{0}, …, x_{n−1}) ↦ ∃zφ(z, ẋ_{0}, …, ẋ_{n−1})

for φ ∈ Δ_{0}. We fix a truth predicate for all Σ_{1}-formulae, i.e., a Σ_{1}-formula T (f) such that

iΣ_{1} ⊢ T (f) ↔ ‘the Σ_{1}-formula coded by f is true’ .

For more information see [HP93] I.1.d. In the sequel, variables f, g, h are intended to range over codes ∃zφ(z, ẋ_{0}, …, ẋ_{n−1}) of Σ_{1}- formulae. In other words, we have a possibility to quantify over all Σ_{1}-predicates. Finally, let ∨ ◯ be a primitive-recursive function such that

∃zφ(z, ẋ_{0}, …, ẋ_{n−1}) ∨ ◯ ∃zψ(z, ẋ_{0}, …, ẋ_{m−1}) =
∃z(φ(z, ẋ_{0}, …, ẋ_{n−1}) ∨ ψ(z, ẋ_{n}, …, ẋ_{n+m−1})) .

Lemma 7.11 iΣ_{1} proves

(i) (T (e) ∨ T (f)) ↔ T (e ∨ ◯f)

(ii) ∃zφ(z, x_{0}, …, x_{n−1}) ↔ T ( ∃zφ(z, ẋ_{0}, …, ẋ_{n−1}) ), for φ ∈ Δ_{0}. □

The next definition is crucial. It is an extended Friedman-translation combined with the negative translation.

Definition 7.12 (Coquand, Hofmann, [CH98]) For every formula φ we define a formula f ⊩ φ (read: f forces φ) with F V (f ⊩ φ) = F V (φ) ∪ {f}.

\begin{aligned}f ⊩ φ & :≡ & φ ∨ T (f) for φ∈ Δ_{0} \\ f ⊩ φ ∧ ψ & :≡ & f ⊩ φ ∧ f ⊩ ψ \\ f ⊩ φ → ψ & :≡ & ∀g[g ⊩ φ → f ∨ ◯g ⊩ ψ] \\ f ⊩ ∀xφ & :≡ & ∀x(f ⊩ φ) \\ f ⊩ φ ∨ ψ & :≡ & f ⊩ ¬(¬φ ∧ ¬ψ) \\ f ⊩ ∃xφ & :≡ & f ⊩ ¬∀x¬φ. \end{aligned}

These are the main properties of the relation ⊩:

Lemma 7.13 iΣ_{1} proves (i) ∀f(f ⊩ φ → ψ) ↔ ∀f(f ⊩ φ → f ⊩ ψ)

(ii) (f ⊩ ∃zφ(z, x_{0}, …, x_{n−1})) ↔ ∃zφ(z, x_{0}, …, x_{n−1}) ∨ T (f), for φ ∈ Δ_{0} whence for ψ ∈ Σ_{1} the formula f ⊩ ψ is still equivalent to a Σ_{1}-formula

(iii) For φ ∈ Φ′_{n} the formula f ⊩ φ is equivalent to a formula in Φ′_{n} (n ⩾ 1).

Proof: (i) is straightforward by the definition. (ii) is the crucial point. By definition f ⊩ ∃zφ(z, x_{0}, …, x_{n−1}) is equivalent to

(∗) ∀g[∀h∀z[φ(z, x_{0}, …, x_{n−1}) ∨ T (h) → T (g ∨ ◯h)] → T (f ∨ ◯g)]

By T (f) → T (f ∨ ◯g) we get the ′ ←′-direction of (ii). Vice versa, we suppose (∗) holds. Let g = ∃zφ(z, ẋ_{0}, …, ẋ_{n−1}) . By 7.11 we have

∀h∀z[(φ(z, x_{0}, …, x_{n−1}) ∨ T (h)) → T (g ∨ ◯h)].

This together with (∗) implies T (f ∨ ◯g), which yields

∃zφ(z, x_{0}, …, x_{n−1}) ∨ T (f).

(iii) This is easy by induction on n using (ii) for n = 1. For the induction step suppose n ⩾ 1 and ψ ∈ Φ′_{n} and χ ∈ Σ_{1}. Then we have again by (ii) and the induction hypothesis:

f ⊩ ∀x(ψ → χ) ↔ \underbrace{∀xg(\underbrace{g ⊩ ψ}_{Φ′_{n}} → \underbrace{f ∨ ◯g ⊩ χ}_{Σ_{1}})}_{Φ′_{n+1}}

This completes the proof of the lemma. □ Remark: If we compare f ⊩ φ with the usual Friedman-translation (2.4), we see that for φ, ψ(x_{0}, …, x_{n−1}) ∈ Δ_{0} we have

((∃xφ)^{−})^{∃zψ(z,x_{0},...,x_{n−1})} ↔ ∃zψ(z, ẋ_{0}, …, ẋ_{n−1}) ⊩ (∃xφ)

For n = 1 this is the theorem by Coquand and Hofmann ([CH98]):

Theorem 7.14 IΠ_{n} ⊢ φ ⇒ iΦ′_{n} ⊢ (f ⊩ φ) (n ⩾ 1).

Proof: Let n ⩾ 1. By Lemma 7.3 (ii) it suffices to show IΦ′_{n} ⊢ φ ⇒ iΦ′_{n} ⊢ (f ⊩ φ). We proceed by induction on the derivations in IΦ′_{n}.

• Logical axioms and rules: Note that the definition of f ⊩ φ includes the negative translation. Therefore this is standard (cf. [Tr73]).

• Non-logical axioms besides induction: These axioms are atomic formulae and hence are forced by f in iΦ′_{n}.

• Induction: We show for all φ ∈ Φ′_{n}

iΦ′_{n} ⊢ f ⊩ [φ(0) ∧ ∀x(φ(x) → φ(x + 1)) → ∀xφ(x)].

By lemma 7.13 (i) it suffices to show

∀f[f ⊩ φ(0) ∧ f ⊩ ∀x(φ(x) → φ(x + 1)) → f ⊩ ∀xφ(x)].

We take an arbitrary f and suppose

(1) f ⊩ φ(0) and (2) f ⊩ ∀x(φ(x) → φ(x + 1)).

(2) is by definition

(3) ∀x∀g[g ⊩ φ(x) → f ∨ ◯g ⊩ φ(x + 1)].

The choice g = f yields

(4) ∀x[f ⊩ φ(x) → f ⊩ φ(x + 1)].

Since (f ⊩ φ(x)) ∈ Φ′_{n} we obtain by (1) and (4) using Φ′_{n} induction

f ⊩ ∀xφ(x)

and we are done. □ Now we are able to prove 7.6: For n = 0 this is well-known (cf. e.g. [Weh96]). Suppose n ⩾ 1 and IΠ_{n} ⊢ ∀x∃yφ(x, y) with φ ∈ Δ_{0}, i.e

IΠ_{n} ⊢ ∃yφ(x, y).

By the theorem above this yields

iΦ′_{n} ⊢ (f ⊩ ∃yφ(x, y)).

Lemma 7.13 (ii) yields

iΦ′_{n} ⊢ ∃yφ(x, y) ∨ T (f).

Substituting ⊥ for f we obtain

iΦ′_{n} ⊢ ∃yφ(x, y).

Thus we have established 7.6 □

7.a.1 Truth-predicates for Φ′_{n}

Definition 7.15 (Truth-predicates for Φ′_{n}) We define truth-predicates T′_{n} for each natural number n. Let Sat_{Δ_{0}}(x) be a truth-definition for Δ_{0}-formulae, cf. e.g. [HP93], I.1.d. It is clear that all properties for Sat_{Δ_{0}} are provable already in iΣ_{1}, in particular for every φ(x_{1} …, x_{n}) ∈ Δ_{0} we have

iΣ_{1} ⊢ Sat_{Δ_{0}}( φ(ẋ_{1}, …, ẋ_{n}) ) ↔ φ(x_{1}, …, x_{n}).

1. T′_{0} := Sat_{Δ_{0}}

2. We define T′_{1} such that for all φ(x) ∈ Δ_{0} we have

T′_{1}( ∃xφ(x) ) ↔ ∃xSat_{Δ_{0}}( φ(ẋ) ).

3. Now suppose T′_{n} is already defined for n ⩾ 1. We define T′_{n+1} such that for all ψ ∈ Φ′_{n} and χ ∈ Σ_{1} we have

T′_{n+1}( ∀x(ψ → χ) ) ↔ ∀x[T′_{n}( ψ(ẋ) ) → T′_{1}( χ(ẋ) )].

This is possible, since it is primitive recursive to decide for a given z whether it is the Gödelnumber of a formula ∀x(ψ → χ) ∈ Φ_{n+1} and further to recover the Gödelnumbers of the sub-formulae ψ, χ from x.

Theorem 7.16 (i) T′_{n} ∈ Φ′_{n}.

(ii) For all φ ∈ Φ′_{n} we have

iΣ_{1} ⊢ T′_{n}( φ(ẋ_{1}, … ẋ_{n}) ) ↔ φ(x_{1}, …, x_{n}).

Proof: Straightforward. □

Corollary 7.17 iΦ′_{n} is finitely axiomatized. □

7.b Fragments iΦ_{n}

The union of all formula classes Φ′_{n} is sufficient to obtain the full Π_{2}-strength of HA. However, not every formula is equivalent in iΔ_{0} to a formula in some Φ′_{n}. Therefore we define

Definition 7.18 (i) Φ_{0} := Δ_{0}

7.b Fragments iΦ_{n}  105

(ii) Φ_{1} := Σ_{1} (iii) Suppose Φ_{n−2}, Φ_{n−1} are defined for n ⩾ 2. Then Φ_{n} consists of all formulae

∀x(ψ → ∃zχ) where ψ ∈ Φ_{n−1}, χ ∈ Φ_{n−2} and x is a finite tuple of variables which may occur both in ψ and χ.

{Lemma 7.19 \atop (i)} Every φ ∈ Φ_{2} is equivalent in iΔ_{0} to a Π_{2}-formula. (ii) For n > 1 every φ ∈ Φ_{n} is equivalent in IΔ_{0} to a formula in Π_{n}. Proof: The first claim is obvious and includes the case n = 2 of the second claim. To prove (ii) we hence may suppose that by induction hypothesis ψ ∈ Π_{n−1} and χ ∈ Π_{n−2}. Then we have with classical logic

∀x(ψ → ∃zχ) ↔ ∀x(\underbrace{¬ψ}_{Σ_{n−1}} ∨ \underbrace{∃zχ}_{Σ_{n−1}})

and this is in Π_{n}. □

Lemma 7.20 Suppose n ⩾ 2. Then we have within iΔ_{0} (by φ ∈ Φ_{n} we mean: φ is in iΔ_{0} provably equivalent to a formula in Φ_{n}):

(i) φ ∈ Φ_{n} ⇒ ∀xφ ∈ Φ_{n} (ii) φ ∈ Φ_{n−2} ⇒ ∃xφ ∈ Φ_{n} (iii) φ_{0}, φ_{1} ∈ Φ_{n−2} ⇒ φ_{0} ∨ φ_{1} ∈ Φ_{n} (iv) φ_{0}, φ_{1} ∈ Φ_{n} ⇒ φ_{0} ∧ φ_{1} ∈ Φ_{n}

(v) φ_{0} ∈ Φ_{n−1}, φ_{1} ∈ Φ_{n} ⇒ φ_{0} → φ_{1} ∈ Φ_{n} Proof: By induction on n. The case n = 2 follows from lemma 7.19 (i). Suppose n > 2. Then (i) is clear and (ii) holds by definition. For (iii) note that

φ_{0} ∨ φ_{1} ↔ ∃z(z = 0 → φ_{0} ∧ z ≠ 0 → φ_{1}).

By induction hypothesis and (ii) this yields the claim. (iv) Suppose φ_{i} ≡ ∀x_{i}(ψ_{i} → ∃zχ_{i}) and ψ_{i} ∈ Φ_{n−1}, χ_{i} ∈ Φ_{n−2} for i = 0, 1. Then the induction hypothesis yields

(φ_{0} ∧ φ_{1}) ↔

∀wx_{0}x_{1}[(w = 0 → ψ_{0} ∧ w ≠ 0 → ψ_{1}) → (w = 0 → ∃zχ_{0} ∧ w ≠ 0 → ∃zχ_{1})] ↔

∀wx_{0}x_{1}[\underbrace{(w = 0 → ψ_{0} ∧ w ≠ 0 → ψ_{1})}_{Φ_{n−1}} → ∃z \underbrace{(w = 0 → χ_{0} ∧ w ≠ 0 → χ_{1})}_{Φ_{n−2}}].

Hence this is equivalent to a formula in Φ_{n}.

(v) Suppose φ_{0} ∈ Φ_{n−1} and φ_{1} ≡ ∀x(ψ → ∃zχ) ∈ Φ_{n} with ψ ∈ Φ_{n−1}, χ ∈ Φ_{n−2}. Whence we have

(φ_{0} → φ_{1}) ↔ (φ_{0} → ∀x(ψ → ∃zχ)) ↔ ∀x((\underbrace{φ_{0} ∧ ψ}_{Φ_{n−1}}) → ∃zχ).

This concludes the proof of the lemma. □ The classes Φ_{n} provide normal-forms for intuitionistic arithmetic:

Corollary 7.21 Every φ ∈ L_{A} is equivalent in iΔ_{0} to a formula φ ∈ Φ_{n} for some n. Hence the formulae in ⋃_{n∈ω} Φ_{n} are normal forms for L_{A}. □

Obviously we have Φ′_{n} ⊂ Φ_{n} and hence have

Corollary 7.22 IΠ_{n} and iΦ_{n} are Π_{2}-equivalent. Furthermore, iΦ_{n} satisfies Markov’s rule for Σ_{1}-predicates.

Definition 7.23 (Truth-predicates for Φ_{n}) It is straightforward to define truth-predicates for the classes Φ_{n}: T_{0} := T′_{0}, T_{1} := T′_{1}. Suppose T_{n} is already defined for n ⩾ 1. Then we define T_{n+1} such that

T_{n+1}( ∀x(ψ → ∃zχ) ) ↔ ∀x[T_{n}( ψ(ẋ) ) → ∃zT_{n−1}( χ(ż, ẋ) )].

Corollary 7.24 (i) T_{n} ∈ Φ_{n}.

(ii) For all φ ∈ Φ_{n} we have

iΣ_{1} ⊢ T_{n}( φ(ẋ_{1}, … ẋ_{n}) ) ↔ φ(x_{1}, …, x_{n}).

7.c Fragments iΦ̃_{n}  107

Proof: Straightforward. □

Corollary 7.25 iΦ_{n} is finitely axiomatized. □

7.c Fragments iΦ̃_{n}

We have ⋃_{n∈ω} Φ_{n} = L_{A} only modulo equivalence in iΔ_{0}. To obtain classes that literally exhaust L_{A} we define

Definition 7.26 (The formula classes Φ̃_{n}) (i) Φ̃_{0} := Δ_{0}

(ii) Φ̃_{1} := Σ_{1}

(iii) Suppose Φ̃_{n}, Φ̃_{n−1} are already defined for n ⩾ 1. We define Φ̃_{n+1} by the following clauses:

1. Φ′_{n+1} ⊆ Φ̃_{n+1}

2. φ, ψ ∈ Φ̃_{n} ⇒ φ ∧ ψ, φ → ψ, φ ∨ ψ, ∀xφ ∈ Φ̃_{n+1} 3. φ ∈ Φ̃_{n−1} ⇒ ∃xφ ∈ Φ̃_{n+1}

Theorem 7.27 (i) IΠ_{n} and iΦ̃_{n} are Π_{2}-equivalent.

(ii) ⋃_{n∈ω} Φ̃_{n} = L_{A}

Proof: (i) is straightforward: for n ⩾ 2 classically every formula in Φ̃_{n} is equivalent to a formula in Π_{n}. Furthermore we have Φ′_{n} ⊆ Φ̃_{n}. (ii) is obvious. □

Corollary 7.28 iΦ̃_{n} is closed under Markov’s rule for Σ_{1}-predicates, i.e

iΦ̃_{n} ⊢ ¬¬∃xφ(x) ⇒ iΦ̃n ⊢ ∃xφ(x).

hold for all φ ∈ Δ_{0}. □

Definition 7.29 (Truth-predicates for Φ̃_{n}) Using T′_{n} we can easily define truth-predicates T̃_{n} for the formula classes Φ̃_{n}: T̃_{0} := T′_{0}, T̃_{1} := T′_{1}. Suppose T̃_{n} is already defined for n ⩾ 1. Then we define T̃_{n+1} by definition by cases such that

1. For φ ∈ Φ′_{n+1}

T̃_{n+1}( φ ) ↔ T′_{n+1}( φ )

2. For φ, ψ ∈ Φ̃_{n}

T̃_{n+1}( φ ∧ ψ ) ↔ T̃_{n}( φ ) ∧ T̃_{n}( ψ )

and similar all remaining cases according to the definition of Φ̃_{n+1}.

3. For φ ∈ Φ̃_{n−1}

T̃_{n+1}( ∃xφ ) ↔ ∃xT̃_{n−1}( φ(ẋ) ).

Theorem 7.30 For all φ ∈ Φ̃_{n} we have iΣ_{1} ⊢ T̃_{n}( φ(ẋ_{1}, … ẋ_{n}) ) ↔ φ(x_{1}, …, x_{n}). □

Corollary 7.31 iΦ̃_{n} is finitely axiomatized. □

7.c.1 More on Π_{2}-conservativity results

In this subsection we discuss two more ways to obtain Π_{2}-conservativity results. In the preliminaries we mention that the standard way to prove Π_{2}-conservativity of a classical theory over its intuitionistic counterpart is via negative- and Friedman-translation. This method applies for example to show the Π_{2}-conservativity of PA over HA and of IΔ_{0} over iΔ_{0}. However, it depends heavily on some special properties of HA and iΔ_{0} (in both cases quite different reasons). In many other cases this technique does not apply, e.g., not for IΣ_{1} and IΠ_{n+2} (although they are Π_{2}-conservative over iΣ_{1} and iΦ_{n+2} resp).

Given formula classes Γ_{1}, Γ_{2} the following condition is sufficient to make the Friedmantranslation work. Provided

φ ∈ Γ_{1} and ψ ∈ Σ_{1} ⇒ (φ−)^{ψ} ∈ Γ_{2},

then IΓ_{1} is Π_{2}-conservative over iΓ_{2}. Note that PA and HA trivially fulfill these conditions, but IΔ_{0} and iΔ_{0} a priori not, here one uses lemma 2.5.

For the hierarchy Φ′_{n} of the previous section, this condition is not satisfied: for φ in Π_{2} and ψ in Σ_{1} we do not have (φ^{−})^{ψ} ∈ Π_{2} (recall: Π_{2} = Φ′_{2} in iΔ_{0}). The following example shows how we can modify the definition of Φ′_{n} to fulfill the condition. Of course, the modified class is somehow artificial. However, this example is more intended to indicate the limits of the Friedman-translation.

Definition 7.32 (The formula-classes Φ_{n}^{∗}) We define classes Φ_{n}^{∗} of L_{A}-formulae:

(i) Φ_{0}^{∗} := Δ_{0} (ii) Φ_{1}^{∗} := Σ_{1} (iii) Φ_{2}^{∗} := {(φ^{−})^{ψ} | φ ∈ Π_{2}, ψ ∈ Σ_{1}} (iv) Suppose Φ_{n}^{∗} is already defined for n ⩾ 2. We inductively define Φ_{n+1}^{∗}

1. Φ_{n}^{∗} ⊆ Φ_{n+1}^{∗}

2. φ ∈ Φ_{n}^{∗}, ψ ∈ Φ_{n+1}^{∗} ⇒ φ → ψ ∈ Φ_{n+1}^{∗}

3. φ ∈ Φ_{n+1}^{∗} ⇒ ∀xφ(x) ∈ Φ_{n+1}^{∗}

Lemma 7.33 These are the basic properties of Φ_{n}^{∗} (i) Every φ ∈ Φ_{n}^{∗} is classically equivalent to a Π_{n}-formula (n ⩾ 2) (ii) If φ ∈ Φ_{n}^{∗} and ψ ∈ Σ_{1} then (φ^{−})^{ψ} is in Φ_{n}^{∗} (n ⩾ 2) (iii) IΠ_{n} = IΦ_{n}^{∗}. Proof: (i) is straightforward. (ii) For n = 2 this holds by definition. For n > 2 this is straightforward by induction hypothesis, since the definition of Φ_{n}^{∗} for n ⩾ 3 involves only negative connectives: they commute with the negative- and the Friedman-translation. (iii) is obvious. □

Theorem 7.34 For n ⩾ 2, all formulae φ ∈ L_{A} and ψ ∈ Σ_{1} we have

IΦ_{n}^{∗} ⊢ φ ⇒ iΦ_{n}^{∗} ⊢ (φ^{−})^{ψ}. Proof: This is similar to the proof of theorem 2.6 and uses 7.33 (ii). □

Corollary 7.35 IΠ_{n} is Π_{2}-conservative over iΦ_{n}^{∗}. Proof: For n = 0, 1 this is clear. Suppose n ⩾ 2, φ ∈ Δ_{0} and IΠ_{n} ⊢ ∀x∃yφ(x, y), hence be inversion and lemma 7.33 (iii)

IΦ_{n}^{∗} ⊢ ∃yφ(x, y).

The previous theorem yields

iΦ_{n}^{∗} ⊢ (¬¬∃yφ(x, y))^{∃yφ(x,y)}.

This is by lemma 2.5

iΦ_{n}^{∗} ⊢ [∃y(φ(x, y) ∨ ∃yφ(x, y)) → ∃yφ(x, y)] → ∃yφ(x, y).

The premise […] is derivable in intuitionistic predicate logic, hence we get by modus ponens

iΦ_{n}^{∗} ⊢ ∃yφ(x, y).

This concludes the proof of the corollary. □

The following is a more direct strategy to show that a classical theory T is Π_{2}-conservative over an intuitionistic system S. We find a family of recursive functions (g_{k})_{k∈ω} with the following property: For every provably recursive function f of T there is a natural number k such that g_{k} majorizes f. To prove the Π_{2}-conservativity of T over S, now it is sufficient to show that T is Π_{1}-conservative over S and that all functions g_{k} are provably total in S.

Theorem 7.36 Given a classical L_{A}-theory T and an intuitionistic L_{A}-theory S. Suppose further that there are recursive functions (g_{k})_{k∈ω} with Σ_{1}-graphs G_{g_{k}} such that the following three conditions are fulfilled:

(i) T is Π_{1}-conservative over S.

(ii) For any Π_{2}-theorem ∀x∃yφ(x, y) of T exists k ∈ ℕ with

T ⊢ ∀xz[G_{g_{k}}(x, z) → ∃y ⩽ z φ(x, y)].

(iii) S ⊢ ∀x∃yG_{g_{k}}(x, y) for all k ∈ ℕ.

Then T is Π_{2}-conservative above S.

Proof: Suppose T ⊢ ∀x∃yφ(x, y) where φ ∈ Δ_{0}. Then there is by (ii) k ∈ ℕ such that

(1) T ⊢ ∀xz[G_{g_{k}}(x, z) → ∃y ⩽ z φ(x, y)].

The Π_{1}-conservativity gives (note that G_{g_{k}} occurs negative)

(2) S ⊢ ∀xz[G_{g_{k}}(x, z) → ∃y ⩽ zφ (x, y)].

7.c Fragments iΦ̃_{n}  111

We show S ⊢ ∀x∃y φ(x, y). Let an arbitrary x be given. By (iii) there is a z such that G_{g_{k}}(x, z). By (2) follows ∃y ⩽ zφ(x, y). Hence S ⊢ ∀x∃y φ(x, y). □

If we try to apply this strategy to IΠ_{n}, we can improve our artificial definition of Φ_{2}^{∗} in the following way:

Definition 7.37 (The formula-classes Φ_{n}^{⋄}) We define classes Φ_{n}^{⋄} of L_{A}-formulae:

(i) Φ_{0}^{⋄} := Δ_{0} (ii) Φ_{1}^{⋄} := Σ_{1} (iii) Φ_{2}^{⋄} := Π_{2} ∪ {φ^{−} | φ ∈ Π_{2}} (iv) Suppose Φ_{n}^{⋄} is already defined for n ⩾ 2. We inductively define Φ_{n+1}^{⋄}

1. Φ_{n}^{⋄} ⊆ Φ_{n+1}^{⋄}

2. φ ∈ Φ_{n}^{⋄}, ψ ∈ Φ_{n+1}^{⋄} ⇒ φ → ψ ∈ Φ_{n+1}^{⋄}

3. φ ∈ Φ_{n+1}^{⋄} ⇒ ∀xφ(x) ∈ Φ_{n+1}^{⋄} Now we apply theorem 7.36 to T := IΠ_{n} and S := iΦ_{n}^{⋄} for n ⩾ 2. The family of functions

that we use is (f_{ω_{n+1}^{k}})_{k∈ω}, where for any α < ε_{0}

f_{α}(x) = min{y | (… ((α[x])[x + 1]) …)[y] = 0}; here ·[·] : ε_{0} × ω → ε_{0} denotes the following fundamental-sequences for ε_{0}:

(i) 0[n] := 0; (α + 1)[n] := α (ii) (α + λ)[n] := α + λ[n] (iii) ω^{α+1}[n] := ω^{α} · (n + 2) (iv) ω^{λ}[n] := ω^{λ[n]+1}. The functions f_{α} have the following properties:

(i) f_{α}(x) = f_{α[x]}(x + 1), hence (ii) f_{ω^{λ}}(x) = f_{ω^{λ[x]+1}}(x + 1) and f_{ω^{α+1}}(x) = f_{ω^{α}·(x+2)}(x + 1) (iii) f_{ω^{α_{1}}+···+ω^{αn}}(x) = f_{ω^{α_{1}}}(… (f_{ω^{αn}}(x) …)) for α_{1} ⩾ … ⩾ α_{n}.

It is clear that all these notions are L_{A}-definable and that iΣ_{1} proves their basic properties. We use Greek letters α, β, γ, … for variables ranging over codes of ordinals less than ε_{0}. We define ω_{0}^{α} := α; ω_{n+1}^{α} := ω^{(ω_{n}^{α})}.

Definition 7.38 Suppose φ(x) ∈ L_{A} and ≺ is the standard Δ_{0}-definable binary less-than-relation on codes of ordinals less than ε_{0}. We define

Prog(φ(x)) :≡ ∀α[∀β ≺ α φ(β) → φ(α)]
TI(α, φ(x)) :≡ Prog(φ(x)) → ∀β ≺ α φ(β)
φ(α)′  :≡ ∀γ [∀β ≺ γ φ(β) → (∀β ≺ γ + ω^{α}) φ(β)]

The iterated jump-operator is defined by φ^{(0)} :≡ φ; φ^{(n+1)} :≡ (φ^{(n)})′.

Claim 1: IΠ_{n} is Π_{1}-conservative over iΦ_{n}^{⋄}. Proof: This is proven via the negative translation. Note that due to the definition of Φ_{2}^{⋄} we have φ ∈ Π_{n} ⇒ φ^{−} ∈ Φ_{n}^{⋄}.

Claim 2: Suppose n > 0 and IΠ_{n} ⊢ ∀x∃yφ(x, y) and φ ∈ Δ_{0}. Then there is k ∈ ℕ with

IΠ_{n} ⊢ ∀xy[f_{ω_{n}^{k}}(x) = y → ∃w ⩽ y φ(x, w)].

Proof (cf. [HP93], page 248, theorem 3.10 (iv)): We define ψ(x, y) :≡ φ(x, y) ∧ ∀z < y¬φ(x, z). Then still IΠ_{n} ⊢ ∀x∃yψ(x, y). For a contradiction we assume that

T_{∞} :≡ IΠ_{n} + {∃xyz[ψ(x, y) ∧ f_{ω_{n}^{k}}(x) = z ∧ y ⩾ z] | k ∈ ℕ}

is a consistent theory with a nonstandard-model M. An overspill-argument (cf. [Ka91], page 70) yields an nonstandard-element c ∈ M such that

∀u ⩽ c(∃xyz[ψ(x, y) ∧ f_{ω_{n}^{u}}(x) = z ∧ y ⩾ z]).

Therefore exist a, b ∈ M such that

M ⊧ (… ((ω_{n}^{c}[a])[a + 1]) …)[b] = 0.

The proof of lemma 11.1 of [AS97] gives a cut I ⊆_{e} M such that I ⊧ IΠ_{n} and a ∈ I < b. Since M ⊧ φ(a, b) ∧ ∀w < b¬φ(a, w) we obtain the contradiction I ⊧ ¬∃yφ(a, y). Hence T_{∞} is inconsistent and we obtain k ∈ ℕ which fulfills the claim. □

Claim 3: iΦ′_{n} ⊢ ∀x∃y f_{ω_{n+1}^{k}}(x) = y for all k ∈ ℕ Proof: For n = 0 this is obvious. For n = 1 note that all f_{ω^{k}} are primitive-recursive. Therefore suppose n > 1. We first show for φ(α) :≡ ∀x∃y f_{ω^{α}}(x) = y:

(1) iΦ′_{2} ⊢ Prog(φ(α)).

Suppose α is given such that:

(2) ∀β ≺ α [∀x∃yf_{ω^{β}}(x) = y]

To prove ∀x∃yf_{ω^{α}}(x) = y we consider two cases:

1. α ∈ Lim. Then f_{ω^{α}} = f_{ω^{α[x]+1}}(x + 1). By α[x] + 1 ≺ α there is by (2) y := f_{ω^{α[x]+1}}(x + 1) and therefore f_{ω^{α}}(x) = y.

2. α = α_{0} + 1. Then f_{ω^{α}}(x) = f_{ω^{α_{0}}·(x+2)}(x + 1). By α_{0} ≺ α and (2) we get

∀w∃zf_{ω^{α_{0}}}(w) = z.

By Π_{2}-induction on x one shows

(3) ∀w∃zf_{ω^{α_{0}}·(x+2)}(w),

using property (iii) for the functions f_{γ}. This yields (1). We further have for ψ ∈ Φ′_{n} : iΦ′_{n} ⊢ Prog(ψ) → Prog(ψ′) and hence:

(4) iΦ′_{n} ⊢ TI(α, ψ′) → TI(ω^{α}, ψ).

For this proof see lemma 15.6 on page 73 of [Po89] and note that the proof given there is constructive. Now we easily check that φ^{(n−2)} is in Φ′_{n}. For any k ≺ ω we have

⊢ TI(k, φ^{(n−1)}).

n − 1 applications of (4) give

⊢ TI(ω_{n−1}^{k}, φ).

(1) yields

⊢ ∀β ≺ ω_{n}^{k} [∀x∃yf_{ω^{β}}(x) = y].

One final application of (1) gives

⊢ ∀x∃y f_{ω_{n}^{k}}(x) = y.

This shows that any f_{ω_{n}^{k}} is iΦ′_{n} provably recursive and completes the proof of claim 3. From theorem 7.36 and claims 1-3 we obtain (for claim 3 note Φ′_{n} ⊆ Φ_{n}^{⋄}):

Corollary 7.39 IΠ_{n} is Π_{2}-conservative over iΦ_{n}^{⋄}. □

7.d Fragments iΘ_{n}

The theorem by Visser and Wehmeier (7.58) shows that prenex induction is weak in the context of intuitionistic arithmetic. In this section we extend this result: For n ⩾ 2 we close Φ_{n} also under existential quantification (this class we call Θ_{n}) and show that still iΘ_{n} and IΠ_{n} are Π_{2}-equivalent.

Definition 7.40 (Formula-classes Θ_{n}) (i) Θ_{0} := Δ_{0}

(ii) Θ_{1} := Σ_{1}

(iii) Suppose Θ_{n} is already defined for n ⩾ 1. We inductively define Θ_{n+1} 1. Θ_{n} ⊆ Θ_{n+1}

2. φ, ψ ∈ Θ_{n+1} ⇒ φ ∧ ψ, φ ∨ ψ, ∃xφ(x), ∀xφ(x) ∈ Θ_{n+1} 3. φ ∈ Θ_{n}, ψ ∈ Θ_{n+1} ⇒ φ → ψ ∈ Θ_{n+1}

We immediately verify

Lemma 7.41 Φ_{n} ⊆ Θ_{n}. □

Hence we have

Corollary 7.42 IΠ_{n} is Π_{2}-conservative over iΘ_{n}. □

It remains to prove the converse of corollary 7.42. We obtain this result by a functional interpretation due to Parsons of iΘ_{n+1} in fragments T_{n} of Gödel’s T. We need a series of definitions:

Definition 7.43 (Theories T_{n} of primitive-recursive functionals of finite type) We now define fragments T_{n} of Gödel’s T. The only difference to T is that we have recursors R_{σ} only for rk(σ) ⩽ n:

R_{τ} : τ → (o → τ → τ) → o → τ for τ with rk(τ) ⩽ n,

and case distinction functionals

D_{σ} : o → σ → σ → σ for arbitrary σ.

In T the case distinction functionals are definable using R_{σ}, but since T_{n} contains recursors only of type σ with rk(σ) ⩽ n, we need to add D_{σ}.

7.d Fragments iΘ_{n}  115

Lemma 7.44 (Some useful functionals definable in T_{0}, simultaneous recursion in T_{n}) T_{0} allows us to define λ-abstraction using the combinators K and S, i.e. for any a : τ there is a term λx^{σ}.a : σ → τ with x^{σ} ∉ λx^{σ}.a and T_{0} proves

(λx^{σ}.a)b = a[x^{σ} := b] and
b = c → λx^{σ}.(a[b]) = λx^{σ}.(a[c]) (x^{σ} ∉ b, c).

In T_{0} we can define all primitive-recursive functions and prove their defining equations, i.e we have primitive-recursive functionals +, ·, max, min, · −, | · − · |, the characteristic function for the relation <, a predecessor functional δ, an iterator J_{o} of type o → o → o, a bounded μ-operator and an iterated maximum-operator Max. This allows us to define characteristic terms for formulae of L(T_{0}) of type o. Further we have an elementary pairing function ⟨·, ·⟩ with projections j_{l}, j_{r} which can easily be generalized to iterated n-ary pairing-functions and projections (n ∈ ω). □

Lemma 7.45 (Simultaneous Recursion in T_{0}) Given terms a_{1}, …, a_{n} : o ∈ T_{0} and b_{i} : o → \underbrace{o → … → o}_{n} → o ∈ T_{0} (i = 1, …, n). Then

there are terms f_{i} : o → o ∈ T_{0} with

f_{i}0 = a_{i}, f_{i} suc t = b_{i}t(f_{1}t) … (f_{n}t) i = 1, …, n. □

In T_{n} we do not have pairing for arbitrary types, but [DS70] yields:

Theorem 7.46 (Simultaneous recursion in T_{n} for n ⩾ 1) For terms a_{i}, b_{i} : τ_{i}, o → τ_{1} → … → τ_{m} → τ_{i} ∈ T_{n}(i = 1, …, m) with max_{1⩽i⩽m}{rk(τ_{i})} ⩽ n, there are terms f_{i} : o → τ_{i} ∈ T_{n} with

f_{i}0 = a_{i}, f_{i} suc t = b_{i}t(f_{1}t) … (f_{m}t) i = 1, …, m.

Proof: An inspection of the proof given in [DS70] shows that the used recursors are available in T_{n}. Note that for n ⩾ 1 and rk(τ) ⩽ n we have rk(o → τ) = max{rk(o) + 1, rk(τ)} ⩽ n.□

Lemma 7.47 In T_{0} we can define characteristic terms for Δ_{0}-formulae of L_{A}. □

Definition 7.48 (The Dialectica-translation) We now give a slightly modified Dialectica-translation (but still write φ^{D}). We assign inductively a formula ∃v∀wφ̄ with φ̄ ∈ L(T_{0}) to every formula φ ∈ L_{A} (v, w of arbitrary finite type and F V (φ) = F V (∃v∀wφ̄)):

(i) φ^{D} :≡ χ_{φ} = 0 for φ ∈ Δ_{0}, χ_{φ} the characteristic term of φ.

(ii) The rest remains the same.

A formula φ ∈ L_{A} with φ^{D} ≡ ∃u∀xφ̄ is called ^{D}-interpretable in T_{n}, if there exist terms t ∈ T_{n} containing at most the free variables of φ such that T_{n} ⊢ φ̄[t, w]. Note that for φ ∈ L_{A} with translation ∃v∀wφ̄ the matrix φ̄ is a formula of type o: hence there is a characteristic term for φ̄ in T_{0}.

We define φ^{D} ≡ χ_{φ} = 0 for Δ_{0}-formulae since the language L(T) contains no restricted quantifiers. We have HA^{ω∗} ⊢ φ ↔ φ^{D} for any φ ∈ L_{A}.

Definition 7.49 We assign to every arithmetical formula φ ∈ L_{A} with Dialectica-translation φ^{D} ≡ ∃v_{1}^{σ_{1}} … v_{k}^{σ_{k}}∀w_{1}^{τ_{1}} … w_{l}^{τ_{l}} φ̄ the following rank rk_{T} (φ) ∈ ω:

rk_{T} (φ) := max{rk σ_{1}, …, rk σ_{k}, rk τ_{1}, …, rk τ_{l}}.

This lemma indicates that iΘ_{n} is interpretable in T_{n}:

Lemma 7.50 For n ⩾ 1 we have

Θ_{n} ⊆ {φ | rk_{T} (φ) < n}.

Proof: Easy by induction on the definition of Θ_{n}. □ Remark: For n ⩾ 2 also the converse of this lemma holds, but we do not need that result and therefore skip the tedious proof.

Now we recall the functional-interpretation of iΘ_{n+1} in T_{n} due to Parsons. An inspection of Gödel’s functional interpretation yields that all axioms and rules beside the inductionschema can be interpreted using functionals of T_{0}. The interpretability of the induction schema for formulae of Θ_{n+1} in T_{n} follows from the following theorem by Parsons ([Pa72], page 477):

Theorem 7.51 (Parsons) Suppose φ(x) ∈ Θ_{n+1}. Then φ(0) ∧ ∀x(φ(x) → φ(suc x)) → ∀xφ(x) is interpretable in T_{n}.

Proof: The translation of φ is (using Parsons’ notation and suppressing the tuple-notation and partially the free parameters f_{0}, F)

(φ(0) ∧ ∀x(φ(x) → φ(suc x)) → ∀xφ(x))^{D} ≡ ∃ΓGXY Z∀Hgaf_{0}F [φ̄(0, f_{0}, GHga)∧

[φ̄(XHga, Y Hga, H(XHga)(Y Hga)(ZHga)) →
φ̄(suc(XHga), F (XHga)(Y Hga), ZHga)] → φ̄(a, Γa, g)].

Define by simultaneous recursion (Lemma 7.46, note that rk(f_{0}), rk(g) ⩽ n by the assumption φ(x) ∈ Θ_{n+1} and lemma 7.50)

Γ0 := f_{0}; Γ suc t = F t(Γt)

and further

G′Hga0 := g; G′Hga(suc t) = H(a · − suc t)(Γ(a · − suc t))(G′Hgat),
GHga := G′Hgaa

XHga := μw < a{φ̄(w, Γw, G′Hga(a · − w)) ∧ ¬φ̄(suc w, Γ(suc w), G′Hga(a · − suc w))}

Y Hga = Γ(XHga); ZHga = G′Hga(a · − suc(XHga)).

By induction on t one now proves:

φ̄(0, f_{0}, GHga) ∧ [XHga < a → φ̄(XHga, Y Hga, H(XHga)(Y Hga)(ZHga))
→ φ̄(suc(XHga), F (XHga)(Y Hga), ZHga)] →
t ⩽ a → t ⩽ XHga ∧ φ̄(t, Γt, G′Hga(a · − t)).

For t = a this yields the interpretability. □ The interpretation theorem can now be stated:

Theorem 7.52 Every theorem of iΘ_{n+1} is interpretable in T_{n}.

Proof: By induction on the consequences of iΘ_{n+1}.

1. Propositional logic is standard [Tr73].

2. The axioms of PA^{−} and the identity-schema are obviously interpretable.

3. Quantifier-rules and axioms Q1 − Q4: There are two cases which are not standard due to the translation of Δ_{0}-formulae, namely Q3 and Q4 with a Δ_{0}-main formula. We consider for example Q3. Since we treat restricted quantifiers as abbreviations, Q3 reads

(1) φ → (u < t → ψ) ⊢ φ → ∀u < tψ,

with ψ ∈ Δ_{0}. Due to the fact that Δ_{0}-formulae are unchanged by the translation, we have the following phenomenon: While u is a free variable of the premise, it ‘disappears’ in the conclusion. This does not happen if the quantifier is unrestricted: in this case the translation guarantees that the interpreting terms get u as a parameter. Now suppose (1) was the last inference with ψ ∈ Δ_{0} and φ^{D} ≡ ∃v∀wφ̄. We have

(φ → (u < t → ψ))^{D} ≡ ∃W ∀v(φ̄(v, W v) → (χ_{u<t→ψ} = 0))

and

(φ → ∀u < tψ)^{D} ≡ ∃W ∀v(φ̄(v, W v) → χ_{∀u<tψ} = 0).

By induction-hypothesis there are W_{0} with

T_{n} ⊢ φ̄(v, W_{0}v) → (χ_{u<t→ψ} = 0).

Note that W_{0} may contain the free parameter u. We define

s := μu < t(χ_{¬ψ} = 0) and W := W_{0}[u := s]

and obtain

T_{n} ⊢ φ̄(v, W v) → χ_{∀u<tψ} = 0.

4. Induction-schema: The interpretability follows with Parsons theorem (7.51). □ By the following theorem by Parsons, it is possible to show that IΠ_{n+1} proves every

Π_{2}-theorem of iΘ_{n+1}:

Theorem 7.53 (Parsons [Pa72], theorem 6) Suppose φ ∈ L_{A} and the negative translation φ^{−} is interpretable in T_{n}. Then IΠ_{n+1} ⊢ φ. □

Remark: Parsons originally considered the theories IA_{n}^{Π}: classical number theory with symbols and defining equations for all elementary functions and a schema φ(0) ∧ ∀x < a(φ(x) → φ(suc x)) → φ(a) for φ ∈ Π_{n}. Note that IΠ_{n} ⊢ φ(0) ∧ ∀x < a(φ(x) → φ(suc x)) → φ(a) for φ ∈ Π_{n}.

Corollary 7.54 Suppose iΘ_{n+1} ⊢ ∀x∃yφ(x, y) with φ ∈ Δ_{0}. Then IΠ_{n+1} ⊢ ∀x∃yφ(x, y).

Proof: (∃yφ(x, y))^{−} ≡ ¬∀y¬(φ(x, y))^{−}. With iΔ_{0} ⊢ φ^{−} ↔ φ we have iΘ_{n+1} ⊢ ¬∀y¬φ^{−}. By the interpretation theorem hence the negative translation of ∃yφ(x, y) is interpretable in T_{n}. Parsons theorem (7.53) then yields the assertion. □ With 7.42 this yields:

Corollary 7.55 iΘ_{n} and IΠ_{n} prove the same Π_{2}-formulae. □

Corollary 7.56 iΘ_{n} is closed under Markov’s rule for Σ_{1}-predicates, i.e for φ ∈ Δ_{0} holds

iΘ_{n} ⊢ ¬¬∃xφ(x) ⇒ iΘ_{n} ⊢ ∃xφ(x).

Proof: For n = 0 this is clear. Suppose n ⩾ 1 and iΘ_{n} ⊢ ¬¬∃xφ(x). Then 7.53 gives IΠ_{n} ⊢ ∃xφ(x). Corollary 7.55 gives iΘ_{n} ⊢ ∃xφ(x). □

These are easy applications of theorem 7.51:

Corollary 7.57 The provably recursive functions of iΣ_{2} are exactly the primitive-recursive functions.

Proof: Note that for φ ∈ Σ_{2} we have rk_{T} (φ) = 0. Hence iΣ_{2} is interpretable in T_{0}. □

Corollary 7.58 (Visser, Wehmeier) iΔ_{0}+ prenex induction is Π_{2}-conservative over iΠ_{2}.

Proof: Every prenex formula belongs to Θ_{2} and hence iΔ_{0}+ prenex induction is contained in iΘ_{2}. This is Π_{2}-conservative over iΠ_{2}. □

7.e Characterization of provably recursive and provably total functions

It is well-known that the provably recursive functions of IΠ_{n+1} are exactly the ≺ ω_{n+2}-descent recursive functions. Therefore 7.7 and 7.55 give:

Corollary 7.59 The provably recursive functions of iΦ_{n+1} and iΘ_{n+1} are exactly the ≺ ω_{n+2}-descent-recursive functions. □

Theorem 7.60 (Parsons) The type-1-functionals of T_{n} are ≺ ω_{n+2}-descent-recursive functions. □

This theorem is implicit in [Pa72]. It also follows from a result by Weiermann ([Wei96], corollary 2), where he gives sharp upper bounds for the derivation-length of reductions of terms in T_{n}. We use this result to characterize the provably total functions of iΦ_{n+1} and iΘ_{n+1}.

Theorem 7.61 Let S be either iΘ_{n+1} or iΦ_{n+1}. Suppose S ⊢ ∀x∃yφ(x, y) with φ ∈ L_{A} arbitrary. Then there is a ≺ ω_{n+2}-descent-recursive function f with

HA^{ω∗} ⊢ φ(x, fx).

Proof: Since (∃yφ)^{D} ≡ ∃yv∀wφ̄(x, y, v, w) the interpretation theorem applied to the assumption gives functionals f_{0}, v[x] ∈ T_{n} with T_{n} ⊢ φ̄(x, f_{0}, v, w). Hence for f :≡ λx.f_{0} we have HA^{ω∗} ⊢ φ(x, fx). By theorem 7.60 f is ≺ ω_{n+2}-descent-recursive. □

iΔ_{0} + Collection principles

8.a iΔ_{0} + {Bφ | φ ∈ L_{A}}

The collection-schema for a formula φ ∈ L_{A} is

Bφ :≡ ∀x ⩽ a∃yφ(x, y) → ∃z∀x ⩽ a∃y ⩽ zφ(x, y).

In this section we show that iΔ_{0} + {Bφ | φ ∈ L_{A}} is interpretable in a weak fragment of T_{0}. This system does not contain recursors R_{o} but a Maximum-functional and functionals that are strong enough to interpret the logical rules and axioms. Using a result by Kohlenbach this shows that both the provably recursive and the provably total functions of this theory are bounded by polynomials.

Definition 8.1 (The fragment T^{−} of T_{0}) This is the system T^{−}: We have countably many variables of any type σ and constants suc and K_{στ}, S_{σρτ}, D_{σ} for all types and the corresponding defining equations. Furthermore, we have the following constants of the denoted type:

+ : o → o → o; · : o → o → o; δ : o → o, sg : o → o; · − : o → o → o;
| − | : o → o → o; max : o → o → o; min : o → o → o, μ_{b} : 1 → o → o;
Max : 1 → o → o,

and the following defining equations:

(i) x + 0 = x; x + suc y = suc(x + y)

122  iΔ_{0} + Collection principles

(ii) x · 0 = 0; x · suc y = x · y + x (iii) δ 0 = 0; δ suc x = x (iv) sg 0 = 0; sg suc x = 1

(v) x · − 0 = x; x · − suc y = δ(x · − y) (vi) |x − y| = x · − y + y · − x (vii) max xy = x + |x − y| (viii) min xy = x · − |x − y| (ix) μ_{b}f0 = 0; μ_{b}f suc x = μ_{b}fx + sg(1 · − f(μ_{b}(x)))

(x) Max f0 = 0; Max f suc x = max(Max fx)(fx). Logical axioms and rules of T^{−} are those of T_{0}. We further have the induction rule for all formulae of T^{−}. The system iΔ_{0}^{ω}[Max] is T^{−} with quantifiers for variables of arbitrary types but induction only for formulae φ ∈ L(T^{−}). We further define iΔ_{0}^{ω∗}[Max] := iΔ_{0}^{ω}[Max] + (AC) + (IP) + (MP). The interpretation theorem for iΔ_{0} + {Bφ | φ ∈ L_{A}} can be stated as follows:

Theorem 8.2 Suppose iΔ_{0} + {Bψ | ψ ∈ L_{A}} ⊢ φ(a), φ^{D} ≡ ∃v∀wφ̄. Then there are terms t[a] with

T^{−} ⊢ φ̄[t, w, a]. Proof: The logical axioms and rules and the equality-rule are interpretable using the terms in T^{−}. The induction rule for Δ_{0}-formulae is obviously interpretable in T^{−}. To interpret the induction schema, hence it suffices to show that we can derive the Δ_{0}-induction schema with the rule. This is easy, first show for φ ∈ Δ_{0}:

(∗) φ(0) ∧ ∀y < a(φ(y) → φ(y + 1)) → φ(a) using the induction rule (note: (*) is still in Δ_{0}). Weakening then gives the induction schema for φ. It remains to treat the collection schema. Suppose φ(x, y)^{D} ≡ ∃v∀wφ̄. Then we have

(Bφ)^{D} ≡ ∃X_{0}W_{0}ZY_{1}V_{1}∀Y_{0}V_{0}x_{1}w_{1} \underline{8.a iΔ_{0} + \{Bφ | φ ∈ L_{A}\} 123}

[X_{0}Y_{0}V_{0}x_{1}w_{1} ⩽ a → φ̄(V_{0}(X_{0}Y_{0}V_{0}x_{1}w_{1}), W_{0}Y_{0}V_{0}x_{1}w_{1}, X_{0}Y_{0}V_{0}x_{1}w_{1}, Y_{0}(X_{0}Y_{0}V_{0}x_{1}w_{1}))

→ (x_{1} ⩽ a → (Y_{1}Y_{0}V_{0}x_{1} ⩽ ZY_{0}V_{0} ∧ φ̄(V_{1}Y_{0}V_{0}x_{1}, w_{1}, x_{1}, Y_{1}Y_{0}V_{0}x_{1})))].

We define

X_{0}Y_{0}V_{0}x_{1}w_{1} :≡ x_{1}; W_{0}Y_{0}V_{0}x_{1}w_{1} :≡ w_{1};
Y_{1}Y_{0}V_{0}x_{1} :≡ Y_{0}x_{1}; ZY_{0}V_{0} :≡ (Max Y_{0}a); V_{1}Y_{0}V_{0}x_{1} :≡ V_{0}x_{1}.

This choice gives:

[x_{1} ⩽ a → φ̄(V_{0}x_{1}, w_{1}, x_{1}, Y_{0}x_{1})] → [x_{1} ⩽ a → Y_{0}x_{1} ⩽ (Max Y_{0}a) ∧ φ̄(V_{0}x_{1}, w_{1}, x_{1}, Y_{0}x_{1})],

and completes the proof. □ A result by Kohlenbach shows that the type-1-functionals of T^{−} are polynomially bounded:

Theorem 8.3 (Kohlenbach) For any closed term t ∈ T^{−} of type 1 there is a polynomial p(x) with

iΔ_{0}^{ω}[Max] ⊢ ∀x tx ⩽ p(x).

Proof: The idea of the proof is the following: one introduces the relation s − maj_{ρ} by induction on the type ρ:

- x^{∗}s − maj_{o} x :≡ x^{∗} ⩾ x
- x^{∗}s − maj_{σ→τ} x :≡ ∀y^{∗}^{σ}, y^{σ}(y^{∗}s − maj_{σ} y → x^{∗}y^{∗}s − maj x^{∗}y, xy).

124  iΔ_{0} + Collection principles

One further introduces a fragment T̃ of T^{−} containing only the constants K, S, +, ·, 0, suc. Then one can show that for every closed term t ∈ T^{−} there is t^{∗} ∈ T̃ with t^{∗}s − maj t. If now t ∈ T̃ is a closed term of type 1 one can prove that tx^{o} reduces to a term t̂[x^{o}] without combinators K and S - hence to a polynomial. That means: For t ∈ T^{−} of type 1 there is a polynomial t̂[x] with t̂[x]s − maj_{o} tx. Since s − maj_{0} is the greater-then relation on the natural numbers we are done. For more details see proposition 2.2.29 of [Ko96]. Note that our system iΔ_{0}^{ω}[Max] is related to Kohlenbach’s system G_{2}A_{i}^{ω} (cf. definition 2.2.1 of [Ko96]) with the following differences: iΔ_{0}^{ω}[Max] does not contain R_{ρ}, Φ_{2} and the axiom-group 9) of G_{2}A_{i}^{ω}. Furthermore Kohlenbach’s systems are extensional, and hence the case-distinction-functionals are definable in G_{2}A_{i}^{ω}. It is easy to check that Kohlenbach’s proof can be done in iΔ_{0}^{ω}[Max]. For proposition 2.2.21 of [Ko96] we define strongly majorizing terms for the case-distinction-functionals as follows: Let (with σ ≡ ρ_{1} → … → ρ_{n} → o)

D_{σ}^{∗} :≡ λx^{σ}λy^{σ}λz^{0}.(λw^{ρ_{1}} … λw^{ρ_{n}}.xw⃗ + yw⃗).

Then D_{σ}^{∗} s − maj_{σ→σ→o→o→σ}D_{σ}. □

Corollary 8.4 Suppose φ(x, y) is any formula of L_{A} and iΔ_{0} + {Bψ | ψ ∈ L_{A}} ⊢ ∀x∃yφ(x, y). Then a polynomial p(x) exists with

iΔ_{0}^{ω∗}[Max] ⊢ ∀x∃y ⩽ p(x)φ(x, y).

If φ ∈ Σ_{1} this is already provable in iΔ_{0}^{ω}[Max]:

iΔ_{0}^{ω}[Max] ⊢ ∀x∃y ⩽ p(x)φ(x, y).

Proof: Immediately with the interpretation-theorem 8.2 and Kohlenbach’s theorem 8.3. □

8.b iΔ_{0} + (Coll)

In this we section consider the contrapositive of the collection-schema, i.e., for a formula φ the schema:

Cφ :≡ ∀z∃x ⩽ a∀y ⩽ zφ(x, y) → ∃x ⩽ a∀yφ(x, y).

Cφ follows classical from B¬φ using the law of excluded middle. We show that vice versa iΔ_{0} + (Coll) implies the law of excluded middle, where (Coll) := {Bφ, Cφ | φ ∈ L_{A}}.

Theorem 8.5 iΔ_{0} + (Coll) ⊢ PA.

Proof: Recall that intuitionistic predicate-calculus proves:

(∗) (∃xφ(x) → ψ) ↔ ∀x(φ(x) → ψ)

We define for any formula φ ∈ L_{A} the degree deg(φ) as follows: deg(φ) := 0 for φ ∈ Δ_{0}; deg(φ_{0} j φ_{1}) := max{deg(φ_{0}), deg(φ_{1})}+1 for any binary connective j and deg(Qxφ) := deg(φ) + 1 for both quantifiers Q ≡ ∀, ∃. By induction on deg(φ) we simultaneously prove the following three claims:

\underline{8.b iΔ_{0} + (Coll) 125}

(i) iΔ_{0} + (Coll) ⊢ ¬φ ∨ φ

(ii) Both φ and ¬φ are equivalent (in iΔ_{0} + (Coll)) to prenex formulae of Π_{n} ∪ Σ_{n}.

(iii) ∀w ⩽ sφ, ∀w ⩽ s¬φ, ∃w ⩽ sφ, ∃w ⩽ s¬φ are equivalent (in iΔ_{0} + (Coll)) to prenex formulae of Π_{n} ∪ Σ_{n}.

Then by (i) we have iΔ_{0} + (Coll) = IΔ_{0} + (Coll). The latter theory coincides with PA (cf. [Ka91], p. 83).

1. n = 0: This case is obvious.

2. n ⇝ n + 1:

a) Suppose φ ≡ ∃wφ_{0} with deg(φ_{0}) = n. By I.H. ∃w ⩽ sφ_{0} is equivalent to a formula ψ in Σ_{n} ∪ Π_{n} and hence deg(ψ) ⩽ n. Therefore the I.H. yields

¬∃w ⩽ sφ_{0} ∨ ∃w ⩽ sφ_{0}.

This gives

∀w ⩽ s¬φ_{0} ∨ ∃w ⩽ sφ_{0},

i.e.

∃y ⩽ 1[(y = 0 → ∀w ⩽ s¬φ_{0}) ∧ (y = 1 → ∃w ⩽ sφ_{0})].

It follows

∃y ⩽ 1[(y = 0 → ∀w ⩽ s¬φ_{0}) ∧ (y = 1 → ∃wφ_{0})]

and further

∃y ⩽ 1∀w ⩽ s[(y = 0 → ¬φ_{0}) ∧ (y = 1 → ∃wφ_{0})].

An ∀-introduction yields

∀s∃y ⩽ 1∀w ⩽ s[(y = 0 → ¬φ_{0}) ∧ (y = 1 → ∃wφ_{0})].

The next step is crucial: we apply Cψ where ψ denotes the formula in […]:

∃y ⩽ 1∀w[(y = 0 → ¬φ_{0}) ∧ (y = 1 → ∃wφ_{0})].

This now gives

∃y ⩽ 1[(y = 0 → ∀w¬φ_{0}) ∧ (y = 1 → ∃wφ_{0})]

hence

∀w¬φ_{0} ∨ ∃wφ_{0}.

We now obtain ¬∃wφ_{0} ∨ ∃wφ_{0} and (i) follows. (ii) By I.H. φ_{0} and ¬φ_{0} are in Σ_{n} ∪ Π_{n}, hence ∃wφ_{0} ∈ Σ_{n+1} ∪ Π_{n+1}. With (using (*)) ¬∃wφ_{0} ↔ ∀w¬φ_{0} we also find ¬∃wφ_{0} ∈ Σ_{n+1} ∪ Π_{n+1}. (iii) By (ii) φ and ¬φ are in Σ_{n+1} ∪ Π_{n+1}. Using (Coll) and the I.H. we show (iii) as in the classical case.

b) Suppose now φ ≡ ∀wφ_{0}. By I.H. we obtain ¬∀w ⩽ sφ_{0} ∨ ∀w ⩽ sφ_{0}, hence

¬∀wφ_{0} ∨ ∀w ⩽ sφ_{0}.

Using the same argument as above we infer

¬∀wφ_{0} ∨ ∀wφ_{0}

and get (i). (ii) ∀wφ_{0} ∈ Σ_{n+1} ∪ Π_{n+1} follows immediately by I.H. Now we use

¬∃w ⩽ s¬φ_{0} ∨ ∃w ⩽ s¬φ_{0}

to show (again using an appropriate instance of (Coll))

∀wφ_{0} ∨ ∃w¬φ_{0}.

Hence we have ¬∀wφ_{0} ↔ ∃w¬φ_{0} and the I.H. yields ¬∀wφ_{0} ∈ Σ_{n+1} ∪ Π_{n+1}. (iii) is similar to the case above.

c) The cases φ ≡ φ_{0} ∧ φ_{1} and φ ≡ φ_{0} ∨ φ_{1} follow immediately by I.H.

d) Suppose φ ≡ φ_{0} → φ_{1}. By I.H. for φ_{0}, ¬φ_{0}, φ_{1} and ¬φ_{1} we have

φ ↔ ¬φ_{0} ∨ φ_{1} and ¬φ ↔ φ_{0} ∧ ¬φ_{1}

and we obtain (i), (ii) and (iii). This completes the proof of the theorem. □

Our proof essentially makes use also of instances of Bφ, namely to interchange restricted universal quantifiers with unrestricted existential ones. It is an interesting question what happens when we drop Bφ and ask for iΔ_{0} + {Cφ | φ ∈ L_{A}}. This theory proves at least ¬φ ∨ φ for arbitrary φ ∈ Π_{1} and hence is not contained in HA. It is not clear if it is possible to extend this beyond Π_{1}.

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[Wei96] Weiermann, Andreas: How is it, that infinitary methods can be applied to finitary mathematics? Gödel’s T: a case study, 1996, Journal of Symbolic Logic (to appear).

Glossary

A1 − A11, 7 Q1 − Q4, 8 ∃!yφ(y), 8 L_{∈}, 8 Δ_{0}, 8 CZF , 9 CZF^{−}, 9 (REM), 9 Full, 9 mv, 10 a >−< b, 10 KPω, 10 KPi, 10 KPM, 10 Ad, 10 tran, 10 Δ_{0}, 11 Iφ, 11 iΔ_{0}, 11 PA^{−}, 11 iΓ, 11 HA , 11 IΔ_{0}, 11 IΓ, 11 PA , 11 φ^{−}, 12 φ^{ψ}, 12 T , 12 rk, 12 T, 13 L(T), 13 K_{στ}, 13 S_{ρστ}, 13 R_{τ}, 13 χ_{φ}, 14 φ^{D}, 14 φ_{D}, 14 φ^{S}, 15 φ_{S}, 15 HA^{ω}, 16 HA^{ω+}, 16 L_{A}^{ω}, 16 (AC), 16 (AC_{στ}), 16 (M), 16 (IP), 16 PA^{ω}, 16 HA^{ω∗}, 16 T_{∧}, 17 φ^{∧}, 18 φ_{∧}, 18 (IP_{∧}), 18 (M_{∧}), 18 HA_{0}^{ω}, 19 T_{0}, 19 (E2), 19 maj_{σ}, 19 N − HA^{ω}, 20 I − HA^{ω}, 20 E_{σ}, 20 E − HA^{ω}, 20 WE − HA^{ω}, 21 T_{∈}, 41 ω, 41 Suc, 41 I, 41 N, 41 U, 41 R_{σ}, 41 Δ_{0}, 42 λx^{σ}.a^{τ}, 42 0^{σ}, 44 t_{φ}, 44 χ_{φ}, 46 ↾_{τ}, 46 C_{σ}, 47 (), ()_{0}, ()_{1}, 47 TC, 48 ⟨⟩, 49 lh(s), 49 s^{⌢}⟨x⟩, 49 Finseq a, 49 T_{∈}^{+}, 50 φ^{×} , 53 (∀q^{∈}Q^{≠∅}) , 53 exp, 67 CZF^{ω−}, 68 CZF^{ω}, 68 (WAC_{στ}), 68 (WAC), 68 (IP_{∈}), 68 (M_{∈}), 68 CZF_{∗}^{ω−}, 68 CZF_{∗}^{ω}, 68 L_{∈}^{ω}, 68 L_{∈}^{ω}^{+}, 68 maj_{σ}, 76 T_{∈}^{∗}, 77 R^{+}, 77 ↾^{+}, 77 T_{∈}^{c}, 82 Δ_{0}^{ω}, 82 T_{∈}^{+c}, 82

KPω^{ω}, 83

KPω^{ω}_{∗}, 83

φ^{∨}, 83 +, 88 M, 88 M1 − M2, 88 (Pow), 89 Φ′_{n}, 96 T (f), 100 ∨ ◯, 100 f ⊩ φ, 100 T_{n}, 103 Sat_{Δ_{0}}(x), 103 Φ_{n}, 103 Φ̃_{n}, 106 Φ_{n}^{∗}, 108 Φ_{n}^{⋄}, 110 f_{α}, 110 α[n], 110 Prog, 111 TI, 111 φ′, 111 φ^{n}, 111 Θ_{n}, 113 T_{n}, 113 D_{σ}, 113 J_{o}, 114 χ_{φ}, 114 φ^{D}, 114 φ_{D}, 114 rk_{T} (φ), 115 Bφ, 120 T^{−}, 120 δ, 120 sg, 120 · −, 120 | − |, 120 max, 120 min, 120 μ_{b}, 120 Max, 120 iΔ_{0}^{ω}[Max], 121 iΔ_{0}^{ω∗}[Max], 121 s − maj_{ρ}, 122 Cφ, 123 (Coll), 123 deg(φ), 123

Index

admissible set, 10 almost negative, 11 arithmetic, 11

in all finite types, 16 axiom of choice, 16, 18

weak, 68

Cartesian product, 9 case distinction functionals, 47, 113 characteristic terms, 14, 46, 114 characterization theorem

∨, 84

D, 16
S, 17
×, 70
∧, 19

classical logic, 8 collection principles, 120 combinator, 41 constructive set functionals, 41 constructive set theory, 9

in all finite types, 68 continuous, 17

deduction theorem, 50 definition by cases, 47 Dialectica

interpretation, 15 translation, 14

Diller-Nahm interpretation, 18 translation, 18 distribution lemma, 54

eigenvariable, 8, 9, 43

equality extensional, 20 in higher types, 20, 50, 68 intensional, 20

exponentiation functional, 67 Ext-Rule, 50, 68 extensional, 20 extensionality, 9

finite sequences, 49 finite types, 12 formula

of type o, 14 foundation, 9 Friedman translation, 12 fullness functional, 49, 67 functional interpretation of arithmetic, 12 of constructive set theory, 53 of Kripke-Platek set theories, 82

Gödel’s T, 13

hereditarily majorizable functionals, 19 \underline{Index 139}

set functionals, 76 Heyting arithmetic, 11

in all finite types, 16

independence of premises, 16

variant, 18, 68 induction rule, 14 induction schema, 11 infinity, 9 intensional, 20 interpretation theorem

CZF^{−}, 55 KPM, 88 KPω + (Pow), 89 KPi, 88 CZF_{∗}^{ω−}, 73

KPω^{ω}_{∗}, 84

iΔ_{0} + {Bφ | φ ∈ L_{A}}, 121 iΘ_{n}, 115 HA , 15 HA^{ω}, 18 PA , 15

intuitionistic logic, 7

jump operator, 111

Kripke-Platek set theory, 10

in all finite types, 83 Kuratowski-pair, 9

λ-operator, 43

Markov’s principle, 16

variant, 18, 68 Markov’s rule

for Σ_{1}-predicates, 99, 106, 118

negative translation, 12 normal form, 105

ω-recursion, 48 ordered pairs, 47

pair, 9 Peano arithmetic, 11

in all finite types, 16 predicate logic, 7

classical, 8 intuitionistic, 7

projections, 47 provably recursive functions, 118 provably total functions, 119 provably total Σ-definable set functions

CZF^{−}, 66 CZF_{∗}^{ω−}, 74

KPω^{ω}_{∗}, 87

rank, 12 recursor, 41 relative consistency, 66, 73, 87 restricted excluded middle, 9 restricted quantifiers, 8, 11 restriction functionals, 46

separation, 9 separation terms, 44 set operations, 44 set theory, 8 Shoenfield

interpretation, 16 translation, 15 variant, 83

simultaneous recursion, 48, 114 strong collection, 9 subset collection, 9 \underline{140 Index}

transfinite induction, 111 translation

−, 12

D, 14
S, 15
φ, 12
×, 53
∧, 18
∨, 83

truth predicate, 100, 103, 105, 106 type structure, 13

full, 13 union, 9 weak axiom of choice, 68 ξ-rule, 44 zero-functionals, 44