Vorträge im Sommersemester 2020

April 21st  Farmer Schlutzenberg. Nondefinability of embeddings \(j:V_\lambda\rightarrow V_\lambda\)
Abstract: Assume \(ZF\). We show that there is no limit ordinal \(\lambda\) and
\(\Sigma_1\)elementary \(j:V_\lambda\to V_\lambda\) which is definable
from parameters over \(V_\lambda\). 
April 28th  Matteo Viale (Turin). Tameness for set theory
Abstract: We show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the modeltheoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a \(\Pi_2\)property formalized in an appropriate language for second or third order number theory is forcible from some T extending ZFC + large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T.
Part (but not all) of our results require Schindler and Asperó that Woodin’s axiom \((\ast)\) can be forced by a stationary set preserving forcing. 
May 5th  Andreas Lietz. How to force \( (\ast) \) from less than a supercompact
Abstract: AsperóSchindler have shown that Woodin's axiom \((\ast)\) is a consequence of \(\operatorname{MM}^{++}\) and the latter is known to be forceable from a supercompact cardinal. \((\ast)\) however has consistency strength of \(\omega\)many Woodin cardinals, so it should be possible to force it from a much weaker assumption. We present a construction that does so from strictly less than a \(\kappa^{+++}\)supercompact cardinal \(\kappa\) (+GCH). The strategy will be to iterate the forcing from the proof of \(\operatorname{MM}^{++}\Rightarrow(\ast)\). Two main difficulties arise: Whenever we want to use that forcing we will have to make sure that it is semiproper and that \(\operatorname{NS}_{\omega_1}\) is saturated. We hope that the large cardinal assumption can be lowered to around the region of an inaccessible limit of Woodin cardinals. This is joint work with Ralf Schindler. 
May 12th  Farmer Schlutzenberg. \(j:V_\delta\to V_\delta\) in \(L(V_\delta)\)
Abstract: Assuming \(\mathrm{ZF}+V=L(V_\delta)\) where \(\delta\) is a
limit ordinal of uncountable cofinality, we show there is no
nontrivial \(\Sigma_1\)elementary \(j:V_\delta\to V_\delta\). Reference:
Section 8 of ``Reinhardt cardinals and nondefinability'', arXiv
2002.01215. 
May 19th  Stefan Hoffelner. Forcing the \(\bf{\Sigma^1_3}\)separation property
Abstract: The separation property, introduced in the 1920s, is a classical notion in descriptive set theory. It is wellknown due to Moschovakis, that \(\bf{\Delta^1_2}\)determinacy implies the \(\bf{\Sigma^1_3}\)separation property; yet \(\bf{\Delta^1_2}\)determinacy implies an inner model with a Woodin cardinal. The question whether the \(\bf{\Sigma^1_3}\)separation property is consistent relative to just ZFC remained open however since Mathias’ „Surrealist Landscape“paper from 1968. We show that one can force it over L. 
May 26th  Liuzhen Wu. BPFA and \(\Delta_1\)definablity of \(NS_{\omega_1}\)
UPDATE: This talk starts at 10:15 am
Abstract: I will discuss a proof of the joint consistency of BPFA and \(\Delta_1\)definablity of \(NS_{\omega_1}\). Joint work with Stefan Hoffelner and Ralf Schindler. 
June 9th  Farmer Schlutzenberg. Remarks on rankintorank embeddings
Abstract: Recall that Woodin's large cardinal axiom \(I_0\) gives an ordinal \(\lambda\) and an elementary embedding \(j:L(V_{\lambda+1})\to L(V_{\lambda+1})\) with critical point \(<\lambda\). Using methods due to Woodin, we show that if \(ZFC+I_0\) is consistent then so is \(ZF+DC_\lambda \ +\) "there is an ordinal \(\lambda\) and an elementary \(j:V_{\lambda+2}\to V_{\lambda+2}\)''. (A version with the added assumption that \(V_{\lambda+1}^\#\) exists is due to the author, and Goldberg observed that the appeal to \(V_{\lambda+1}^\#\) could actually be replaced by some further calculations of Woodin's.)
Reference: https://arxiv.org/abs/2006.01077, "On the consistency of \(ZF\) with an elementary embedding from \(V_{\lambda+2}\) into \(V_{\lambda+2}\)". 
June 16th  Farmer Schlutzenberg. Remarks on rankintorank embeddings
Continuation 
June 23rd  Farmer Schlutzenberg. Remarks on rankintorank embeddings
Continuation 
June 30th  Farmer Schlutzenberg. Remarks on rankintorank embeddings
Continuation 
June 7th  Farmer Schlutzenberg. Remarks on rankintorank embeddings
Continuation 
July 14th  Ralf Schindler. \(\mathrm{MM}\) and \( (\ast)^{++}\)
Abstract: The axiom \((\ast)^{++}\) is a strengthening of \((\ast)\) which was
also introduced by Woodin. \((\ast)^{++}\) says that the set of sets of reals,
or equivalently, \(H_{c^+}\), is contained in a \(\mathbb P_{\mathrm{max}}\) extension of a
determinacy model. Woodin showed that \((\ast)^{++}\) is false in all the known
models of Martin's Maximum. We will give a proof of this result. It
remains open if \(\mathrm{MM}^{++}\) refutes \((\ast)^{++}\).