Vorträge im Sommersemester 2021

April 16th  Liang Yu (Nanjing University). A basis theorem for \(\Pi^1_1\)sets.
Abstract: It was claimed by Harrington, but never published, that every nonthin \(\Pi^1_1\)set ranges over an upper cone of hyperarithmetic degrees. We shall give a proof via a full approximation argument.

April 21st  Ralf Schindler. Generation of Grounds.
Abstract: A ground is an inner model of which \(V\) is a generic extension. We will present a general method for producing grounds of a given model of set theory via the HOD direct limit system. This may be used to analyze the \(<\kappa\) mantle of \(L[x]\), \(x\) any real above \(M_1^\sharp\) and \(\kappa\) any cardinal. This is joint work with G. Sargsyan and F. Schlutzenberg. 
April 29th  Farmer Schlutzenberg. Local mantles of \(L[x]\).
UPDATE: This Talk has been moved to Thursday, April 29th at 4:15pm.
Abstract: Recall that for a cardinal \(\kappa\), a \(<\kappa\)ground is an inner model \(W\) of ZFC such that \(V\) is a setgeneric extension of \(W\), as witnessed by a forcing of size \(<\kappa\), and the \(\kappa\)mantle is the intersection of all \(<\kappa\)grounds. We will start with a brief overview
of some known facts on the \(\kappa\)mantle. Following this, assuming sufficient large cardinals, we will analyze the \(\kappa\)mantle \(M\) of \(L[x]\), where \(x\) is a real of sufficiently high complexity, and \(\kappa\) is a limit cardinal of uncountable cofinality in \(L[x]\). We will show in particular that \(M\) models ZFC + GCH + "There is a Woodin cardinal". We will also discuss a variant, joint with John Steel, for the \(\kappa\)cc mantle, where \(\kappa\) is regular uncountable in \(L[x]\) and \(\kappa\leq\) the least Mahlo of \(L[x]\). The proof relies on Woodin's analysis of \(\mathrm{HOD}^L[x,G]\) and Schindler's generation of grounds, and is motivated by work of Fuchs, Sargsyan, Schindler and the author on Varsovian models and the mantle.
References: arXiv:2006.01119, arXiv:2103.12925

May 5th  Jindrich Zapletal (University of Florida). Chromatic numbers of distance graphs on Euclidean spaces.
Abstract: Let \(G_n\) be the graph on \(n\)dimensional Euclidean space connecting points of rational distance. I will show that it is consistent relative to an inaccessible cardinal that ZF+DC holds, chromatic number of \(G_3\) is countable, yet the chromatic number of \(G_4\) is uncountable. I will use the opportunity to explain the basic concepts,methods, and results of geometric set theory, as contained in a recent book with Paul Larson.
In the first lecture, I will provide a broad outline of geometric set theory. I will define balanced forcing, a class of partial orders which can be used to prove numerous independence results in ZF+DC, and prove its central theorems. In its usage and flexibility, balanced forcing is a parallel to proper forcing in the context of choiceless set theory. In the second lecture, I will discuss chromatic numbers of algebraic hypergraphs in general and the rational distance graphs in particular. Finally, I will construct a coloring poset which yields the consistency result mentioned in the first paragraph.

May 12th  Jindrich Zapletal (University of Florida). Chromatic numbers of distance graphs on Euclidean spaces.
Continuation. 
May 19th  Azul Fatalini. Forcing a Mazurkiewicz set.
Abstract: A subset of the plane is called a Mazurkiewicz set iff its intersection with every line is exactly two points. There is a wellknown construction of these sets in ZFC, using transfinite recursion of the length of the continuum. We will talk about the construction of a model of ZF+DC with no wellordering of the reals that has a Mazurkiewicz set. 
June 2nd  Gunter Fuchs (CUNY). Fragments of (diagonal) strong reflection.
INFO: This talk starts at 4:15pm.
Abstract:I will talk about reflection principles that arose out of an attempt to find an analog of Todorcevic's strong reflection principle SRP, which captures many of the major consequences of Martin's Maximum, that works with forcing axioms for other forcing classes, in particular subcomplete forcing. Since SRP fails to encapsulate phenomena of diagonal reflection which follow from MM, I will propose a diagonal version of it that does have these consequences, as well as its fragments. The gist of these principles is that there is a natural strengthening of the concept of a projective stationary set, which I call "spread out", which gives rise to the subcomplete fragments of these strong reflection principles. Part of this work is joint with Sean Cox.

June 9th  Gunter Fuchs (CUNY). Fragments of (diagonal) strong reflection.
INFO: This talk starts at 4:15pm.
Continuation.

June 16th  Stefan Hoffelner. Forcing the \(\Pi^1_3\) Uniformization property.
INFO: This talk starts at 4:15pm.
Abstract: We show how to force the \(\Pi^1_3\) uniformization property over Gödel’s \(L\). With some care, the method can be lifted to canonical inner models with Woodin cardinals, thus producing, for the first time universes in which the \(\Pi^1_{2n}\) uniformization property is true.

June 23rd  Stefan Hoffelner. Forcing the \(\Pi^1_3\) Uniformization property.
INFO: This talk starts at 4:15pm.
Continuation.

June 30th  Diana Montoya (University of Vienna). Independence at uncountable cardinals.
Abstract: In this talk, we will discuss the concept of maximal independent
families for uncountable cardinals. First, we will mention a summary of
results regarding the existence of such families in the case of an
uncountable regular cardinal. Specifically, we will focus on joint work
with Vera Fischer regarding the existence of an indestructible maximal
independent family, which turns out to be indestructible after forcing with
generalized Sacks forcing.
In the second part, we will focus on the singular case and present two
results obtained in joint work with Omer BenNeria. Finally, I will mention
some open questions and future paths of research.

July 8th  Omer BenNeria (Hebrew University, Jerusalem). Mathiastype Criterion for the Magidor Iteration of Prikry forcings.
INFO: This talk starts at 4:15pm.
Abstract: In his seminal work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals known as the Magidor iteration. The purpose of this talk is to present a Mathiastype criterion which characterizes when a sequence of omegasequences is generic for the Magidor iteration.
The result extends a theorem of Fuchs, who introduced a Mathias criterion for discrete products of Prikry forcings. We will present the new criterion, discuss several applications, and outline the main ideas of the proof.

July 14th  Victoria Gitman (CUNY). Characterizing large cardinals via abstract logics.
Abstract: Firstorder logic, the commonly accepted formal system underlying mathematics, must draw however minimally on the properties of the settheoretic universe in which it is defined. Stronger logics such as infinitary logics and secondorder logics require access to much larger chunks of the settheoretic background. Niceness properties of these logics, such as forms of compactness, are naturally connected to the existence of large cardinals. Indeed, many large cardinals can be characterized in terms of compactness properties of strong logics. Strongly compact and weakly compact cardinals \(\kappa\) are precisely the strong and weak compactness cardinals respectively for the infinitary logic \(\mathbb L_{\kappa,\kappa}\). Extendible cardinals \(\kappa\) are precisely the strong compactness cardinals for the infinitary secondorder logic \(\mathbb L^2_{\kappa,\kappa}\). Vopenka's Principle holds if and only if every logic has a strong compactness cardinal. In this talk I will review properties of various logics and how their compactness properties characterize various large cardinals. I will discuss joint work with Will Boney, Stamatis Dimopolous and Menachem Magidor in which we show that the principle \(\mathrm{Ord}\) is subtle, in the presence of global choice, holds if and only if every logic has a stationary class of weak compactness cardinals, i.e., it is the analogue of Vopenka's Principle for weak compactness. We also provide compactness characterizations for various virtual large cardinals using a new notion of a pseudomodel of a theory.