Set Theory Seminar (Winter 2026)

Tuesdays, 16:00-18:00
Organized by Rahman Mohammadpour and Grigor Sargsyan

Updates:

This is an in-person seminar in room 106 at IMPAN in Warsaw

Sebastiano Thei

Title: Another proof of Kunen Inconsistency
Abstract: In our previous seminar, we learned many proofs of Kunen Inconsistency. We also proved that \(\textrm{HOD}\) can be close to \(V\), in the presence of an extendible cardinal. In this talk, we address two related questions:
1. Does the HOD hypothesis imply that \(\textrm{HOD}\) has the cover property above the first strongly compact cardinal?
2. Suppose \(j\) is an elementary embedding from an inner model \(M\) into \(V\). We know that \(M\) cannot be \(V\), but can \(M\) be cardinal correct?
The first question is open, while the second one has a negative answer (Goldberg-T.). Both questions are addressed through the lens of PCF theory.

No talks

Rahman Mohammadpour

Ronnie Chen (University of Warsaw)

Grigor Sargsyan

Lukas Koschat (TU Wien)

Title: Full models of \(\textsf{LSA}\)
Abstract: The main topic of the talk is the following recent result: assuming large cardinals in the region of three supercompact cardinals and letting \(\kappa\) be the least supercompact cardinal, in each forcing extension where \(\kappa\) is countable, the minimal inner model containing all universally Baire sets, \(L(uB)\), contains an full model of \(\textsf{LSA}\) (Largest Suslin Axiom). The Largest Suslin Axiom is a determinacy axiom of high consistency strength, and hence the result implies that in the given context \(L(uB)\) satisfies very strong determinacy axioms. The result is part of joint work with Sandra Müller and Grigor Sargsyan. The talk is aimed at a general set theory audience and thus a major part of the talk will be dedicated to introducing the relevant concepts, and explainingtheir importance to the study of consistency strength, as well as their connections to recent developments in the endeavour of forcing failures of square principles over models of determinacy.

Corentin Lagadec

Title: The strong Chang's conjecture in \(\mathbb{P}_{\textsf{max}}\) models
Abstract: I will present a proof that an improved version of the strong Chang's conjecture can be forced with \(\mathbb{P}_{\textsf{max}}\) over a model of determinacy.

Nam Trang (University of North Texas)

Title: Coherent Sequences Constructed over Models of Determinacy
Abstract: We construct various types of coherent sequences over models of determinacy. We use these sequences and core model induction arguments to calibrate the consistency strength of theories extending \(\textrm{MM}(\mathfrak{c})\) and \(\textrm{CH}\) + there is an \(\omega_1\)-dense ideal on \(\omega_1\) such as (1) \(\textrm{CH}\) + there is an \(\omega_1\)-dense ideal on \(\omega_1 + \neg \square_{\omega_1}\). (2) \(\textrm{CH}\) + there is an \(\omega_1\)-dense ideal on \(\omega_1 + \neg \square(\omega_2)\). This is joint work with M. Zeman.

Jouko Vaananen (University of Helsinki)

Title: New inner models from second order logics
Abstract: I define a new inner model based on replacing first order logic in Godel’s definition of the constructible hierarchy by the fragment of second order logic in which second order variables range over countable subsets of the domain. I compare this inner model to the previously studied inner models from extended logics, mainly \(C^*\) and \(C(aa)\). I will discuss large cardinals in this model and forcing absoluteness of the its theory. There are many open problems. Time permitting, I will also discuss the inner model \(\textrm{HOD}_1\), obtained in a similar way from the existential fragment of second order logic. This is joint work with Menachem Magidor.

Otto Rajala

Title: \(\textrm{GCH}\) in \(C(aa^+)\)
Abstract: I will talk about the inner model \(C(aa^+)\) obtained from the \(aa^{+}\)-quantifier which is a variant of the \(aa\)-quantifier. Hence, \(C(aa^+)\) can be viewed as a variant of the inner model \(C(aa)\) introduced by Kennedy-Magidor-Väänänen. The \(aa^+\)-quantifier says that there is a club of countable subsets of the given structure such that the subformula holds in the next admissible set built from the structure and any one of the subsets. In particular, I will present the proof of \(\textrm{GCH}\) in the model \(C(aa^+)\).

Iljas Farah (York University)

Title: Conjugacy of trivial autohomeomorphisms of \(\beta N\setminus N\).
Abstract: An autohomeomorphism of the Čech--Stone remainder \(\beta N\setminus N\) is called trivial if it has a continuous extension to a map from \(\beta N\) into itself. Such map is determined by an almost permutation, which is a bijection between cofinite subsets of \(N\). By results of W. Rudin and S. Shelah, the question whether nontrivial autohomeomorphisms of \(\beta N\setminus N\) exist is independent from \(\textrm{ZFC}\). We will be considering the so-called rotary autohomeomorphisms. An autohomeomorphism is called rotary if it corresponds to a permutation of \(N\) all of whose cycles are finite. If all autohomeomorphisms are trivial, then the problem of their conjugacy is also trivial (in the usual sense of the word). However the Continuum Hypothesis makes the conjugacy relation nontrivial. While our results are somewhat incomplete, they suffice to decide whether for example the rotary autohomeomorphisms whose cycles have lengths \(2^{2n}\), for \(n\in N\), and \(2^{2n+1}\), for \(n\in N\), are conjugate. This is a joint work with Will Brian.

Piotr Borodulin-Nadzieja (University of Wrocław)

Title: On generalizations of ultrafilters
Abstract: The notion of an ultrafilter can be naturally generalized in different directions, e.g. one can see ultrafilters as particular instances of measures or of Boolean homomorphisms. I will overview some results concerning these generalizations