Talks during the winter of 2022/23

• October 11th - Andreas Lietz. Forcing \(``\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}"\) from large cardinals.

Abstract:  Woodin has developed the \(\mathbb {P}_{\mathrm{max}}\)-variant \(\mathbb Q_{\mathrm{max}}\) to force \(\mathrm{ZFC}+``\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}"\) over canoncical models of determinacy. In practice, the restriction to canonical models of determinacy is an anti large cardinal assumption, so Woodin asked: Assume some large cardinal exists. Is there a semiproper forcing \(\mathbb P\) so that \(V^{\mathbb P}\models``\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}"\)? We answer this question positively.  The strategy is to devise a forcing axiom that, by modifying the Asperó-Schindler argument for \(``\mathrm{MM}^{++}\text{ implies }(*)"\), entails the \(\mathbb Q_{\mathrm{max}}\)-version of Woodin's axiom \( (*)\). We will then force this axiom assuming the existence of a supercompact limit of supercompact cardinals making use of a new iteration theorem.

Talks during the summer of 2022

• April 20th - Farmer Schlutzenberg. The Extent of Determinacy in \(\omega\)-Small Mice.

Abstract: The plan is to discuss some joint work with John Steel,
examining the extent to which determinacy holds in mice. If a mouse
models ZF, then it models choice, so determinacy fails. But it can
model projective determinacy and more, so one can ask how far this
goes. I will discuss some progress complementing earlier work of
Rudominer and Steel on the topic.

• April 27th - Farmer Schlutzenberg. The Extent of Determinacy in \(\omega\)-Small Mice.

Continuation.

• May 4th - Farmer Schlutzenberg. The Extent of Determinacy in \(\omega\)-Small Mice.

Continuation.

• May 11th - Farmer Schlutzenberg. The Extent of Determinacy in \(\omega\)-Small Mice.

Continuation.

• May 18th - Andreas Lietz. More instances of \(MM^{++}\) implies \((*)\)

Abstract: Iterating semiproper forcings to get \(MM^{++}\) and forcing with \(\mathbb P_{\mathrm{max}}\) over \(L(\mathbb R)\) to get (*) are two methods to produce maximal models (in seemingly differing senses of the word "maximal") that turn out to cohere as proven by Asperó-Schindler.

Both axioms forbid the existence of various interesting objects, e.g. Suslin trees. There are many \(\mathbb P_{\mathrm{max}}\) variations that produce a maximal model conditioned to the existence of such objects and few semiproper-style iteration theorems that preserve the existence of such objects. We show that in the examples where both sides of this coin are understood, there usually is a \(MM^{++}\)-style forcing axiom which implies the version of (*) corresponding to the correct \(\mathbb P_{\mathrm{max}}\) variant. Due to the lack of semiproper-style iteration theorems, we also prove a new one out of which we get many instances of \(MM^{++}\) implies (*) uniformly.

• May 25th - Andreas Lietz. More instances of \(MM^{++}\) implies \((*)\).

Continuation.

• June 1st - Andreas Lietz. More instances of \(MM^{++}\) implies \((*)\).

Continuation.

Talks during the winter of 2021/22

.

• October 13th - Farmer Schlutzenberg. Choice principles in local mantles.

  Abstract: The \(\kappa\)-mantle is the intersection of all those forcing
  grounds of \(V\) which can be witnessed with a forcing
  of size \({<\kappa}\). If \(\kappa\) is a strong limit cardinal then the
  \(\kappa\)-mantle models ZF (Usuba), but Choice can fail there (Lietz),
  though if \(\kappa\) is measurable
  then Choice holds (Schindler). In this talk I will discuss some
  further results and open questions on the relationship between choice
  in the \(\kappa\)-mantle and the large cardinal
  properties of \(\kappa\).

• October 20th - Farmer Schlutzenberg. Choice principles in local mantles.

  Continuation.

• October 27th - Andreas Lietz. \((\prod_{i<\kappa} x(i), \leq^*) \) and its Bounding number.

  Abstract: Given an inaccessible cardinal \(\kappa\) and a function \(x\) on \(\kappa\) that is similar to the cardinal-successor-map, we prove two different characterizations of when Hechler forcing below \(x\) adds a \(\kappa\)-dominating function, for example we will show that this happens iff there is a \(\kappa\)-Aronszajn tree with a certain closure property. We will also construct such trees in \(L\) assuming that \(\kappa\) is not weakly compact there. Finally, we present a strategy of separating the bounding number of \((\prod_{i<\kappa} x(i), \leq^*)\) from the \(\kappa\)-bounding number, but are ultimately hampered by the lack of iteration theorems at uncountable cardinals.

• November 10th - Stefan Hoffelner. Modest Forcing Axioms and the \(\Sigma^1_3\)-Uniformization property.

  Abstract: It is well-known that PFA implies PD, hence the \(\Pi^1_{2n+1}\)-uniformization property for every natural number n. We will show that, in contrast, weaker forcing axioms are consistent with the orthogonal behaviour of the uniformization property i.e. there is a model where Martin's Axiom holds and the \(\Sigma^1_3\)-uniformization property is true. We also show that BPFA together with the anti large cardinal assumption \(\omega_1=\omega_1^L\) outright implies that the \(\Sigma^1_3\)-uniformization property holds true.

• November 17th - Stefan Hoffelner. Separating the Reduction from the Uniformization Property.

  Abstract: It is a ZFC result that whenever the Uniformization property holds for a given projective pointclass, then the reduction property must hold for that pointclass. Interestingly, in all known examples of universes where the reduction property holds, it does so because the uniformization property holds there as well. Thus it is natural to ask, whether there are models of ZFC in which the reduction property for some pointclass holds, yet the uniformization property fails. We shall give an outline of how one can produce such a model.

• November 24th - Stefan Hoffelner. Separating the Reduction from the Uniformization Property.

  Continuation.

            UPDATE: This talk has been postponed.

• December 8th - Juan Aguilera. Indiscerinbility Spectra.

Abstract: Let \(A\) be a set of sharps. The spectrum of \(A\) is the set of all countable ordinals which are simultaneously \(a\)-indiscernible for every \(a^\sharp\) in \(A\). We will prove (using large cardinals) that there are sets of sharps with nonempty, but finite spectrum. If time allows, we might prove other related results as well.

• December 15th - Hossein Lamei Ramandi. On the rigidity of Souslin trees and their generic branches.

  Abstract: We show it is consistent that there is a Souslin tree \(S\) such that after forcing with \(S\), \(S\) is Kurepa and for all clubs \(C \subset \omega_1\), \(S\upharpoonright C\) is rigid. This answers a few questions due to Gunter Fuchs. If time permits, we show it is consistent with \(\diamondsuit\) that for every Souslin tree \(T\) there is a dense \(X \subseteq T\) which does not contain a copy of \(T\). This is related to a question due to Baumgartner.

• Dexember 22nd - Stefan Hoffelner. Separating the Reduction from the Uniformization Property (Part 2).

• January 19th - Ben De Bondt. Elevation of \(u_2\) (using elementary submodels as side conditions).

Abstract: We first isolate a particular family of countable elementary submodels of \(H_\theta\) which are characterised using certain closed games by the demand that the second player has a winning strategy. We show that if \(NS_{\omega_1}\) is precipitous, then this set of countable elementary submodels of \(H_\theta\) is projective stationary. Next, we will define and discuss a particular forcing P that consists of finite conditions in which these models feature as side conditions. We will show 1) that this forcing (under suitable conditions) adds a generic iteration of length \(\omega_1\) of an iterable countable transitive model, with direct limit equal to some \(H_\lambda\), and 2) that the properties of the selected models appearing in the forcing’s conditions assure that it is stationary set preserving. The forcing shares these properties with an L-forcing defined by Claverie-Schindler and a Namba-like forcing defined by Ketchersid-Larson-Zapletal, to both of which it shows resemblance. It follows that this side condition approach gives yet another way of increasing the second uniform indiscernible beyond some arbitrary prespecified ordinal. Time allowing, we will discuss how this same method (using other families of countable elementary submodels) can be used (just as L-forcing) to force other \(\Pi_2\)-statements in a stationary set preserving way. This is all joint work with my thesis supervisor Boban Velickovic.

• 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 non-thin \(\Pi^1_1\)-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 set-generic 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 well-known 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 well-ordering 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 Ben-Neria. Finally, I will mention
some open questions and future paths of research.

• July 8th - Omer Ben-Neria (Hebrew University, Jerusalem). Mathias-type 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 Mathias-type criterion which characterizes when a sequence of omega-sequences 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: First-order logic, the commonly accepted formal system underlying mathematics, must draw however minimally on the properties of the set-theoretic universe in which it is defined. Stronger logics such as infinitary logics and second-order logics require access to much larger chunks of the set-theoretic 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 second-order 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 pseudo-model of a theory.

• Novemver 4th - Grigor Sargsyan (Gdansk). Determinacy, forcing axioms and inner models.

Abstract: We will exposit some recent results of the speaker and others
that connect determinacy axioms, forcing axioms and inner models. A
culmination of this work is a recent proof that the most liberally
backgrounded construction of a model build from an extender sequence
cannot be shown to converge in ZFC alone. In this construction, which
is a type of \(K^c\) construction, one uses extenders that are certified by
a Mostowski collapse. This result challenges common perceptions of the
role of the model \(K^c\) in the inner model program.

We will mention a specific consistency result showing that the failure
of \(\square_{\omega_3}\) and \(\square(\omega_3)\) with
\(2^\omega=2^{\omega_1}=\omega_2\) and \(2^{\omega_2}=\omega_3\) is weaker than a
Woodin cardinal that is a limit of Woodin cardinals.

Many people have been involved in this project. The work is heavily
based on the efforts of Steel, Jensen, Woodin, Schindler, Mitchell,
Schimmerling, Trang, Larson, Neeman, Zeman, Schlutzenberg, the speaker
and many others.

• Novemver 11th - Grigor Sargsyan (Gdansk). Determinacy, forcing axioms and inner models.

            Continuation

• Novemver 18th - Grigor Sargsyan (Gdansk). Determinacy, forcing axioms and inner models.

            Continuation

• November 26th - Grigor Sargsyan (Gdansk). Determinacy, forcing axioms and inner models.

  Update: This talk has been moved to thursday 4:15 pm!

            Continuation

• December 2nd -  Paul Larson (Miami Univ., Oxford OH). Square principles in Pmax extensions of Chang models.

  Abstract: We show that the statements \(\square(\omega_{3})\) and \(\square(\omega_{4})\) both fail in the \(\mathbb P_{\mathrm{max}}\) extension of a variation of the Chang model introduced by Sargsyan. This is joint work with Grigor Sargsyan.

• December 9th - Paul Larson (Miami Univ., Oxford OH).  Square principles in Pmax extensions of Chang models.

  Continuation

• December 16th - Paul Larson (Miami Univ., Oxford OH). Square principles in Pmax extensions of Chang models.

            Continuation

• January 13th - Ralf Schindler. The Stable Core.

Abstract: S. Friedman introduced the stable core, \(L[S]\), and showed that \(V\) is class generic over \(L[S]\). He also showed that \(L[S]\) is a
definable inner model of an iterate of a mouse which has a measurable
definable Woodin cardinal. We will produce versions of these results.

• January 20th - Stefan Hoffelner. Forcing the \(\Pi^1_3\) Reduction Property.

  Update: This talk will start at 3:30 pm.

  Abstract: We continue our investigation on the forcability of regularity properties implied by projective determinacy in showing that one can force the \(\Pi^1_3\) Reduction Property over the constructible universe \(L\).

• January 27th - Stefan Hoffelner. Forcing the \(\Pi^1_3\) Reduction Property.

  Continuation

• February 8th - Farmer Schlutzenberg. Full normalization.

  Update: This talk is postponed to 5:00 pm on Monday, February the 8th.

Abstract: We will describe some aspects of full normalization for
stacks of normal iteration trees. More precisely (but somewhat
approximately), given a normal iteration strategy \(\Sigma\) for a fine
structural premouse \(M\), such that \(\Sigma\) and \(M\) both satisfy
appropriate condensation properties, we construct a strategy
\(\Sigma^*\) for transfinite stacks of iteration trees on \(M\), such that
the iterates via \(\Sigma^*\) are in fact iterates via \(\Sigma\). This is
joint work with John Steel.

• February 15th - Farmer Schlutzenberg. Full normalization.

  This talk starts at 5:00 pm.

Continuation.

• February 23rd - Farmer Schlutzenberg. Full normalization.

This talk starts at 5:00 pm.

Continuation.

• March 2nd - Farmer Schlutzenberg. Full normalization.

This talk starts at 5:00 pm.

Continuation.

• April 21st - Farmer Schlutzenberg. Non-definability 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 model-theoretic 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:  Asperó-Schindler 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
  non-trivial \(\Sigma_1\)-elementary \(j:V_\delta\to V_\delta\). Reference:
  Section 8 of ``Reinhardt cardinals and non-definability'', 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 well-known 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 rank-into-rank 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 rank-into-rank embeddings

  Continuation

• June 23rd - Farmer Schlutzenberg. Remarks on rank-into-rank embeddings

  Continuation

• June 30th - Farmer Schlutzenberg. Remarks on rank-into-rank embeddings

  Continuation

• June 7th - Farmer Schlutzenberg. Remarks on rank-into-rank 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 \((\ast)^{++}\).

Talks during the winter of 2019

• October 8th - Farmer Schlutzenberg.  Strategy extensions of \(M_1\)

  Abstract: We discuss the following two questions:
  (1) Let \(x\) be a real of high complexity. Given a cardinal \(\eta\) of \(L[x]\),
  what is the \(\eta\)-mantle of \(L[x]\)? Does it model ZFC?
  (2) Recall that \(M_1\) is the minimal proper class iterable mouse with a Woodin cardinal $\delta$. Woodin showed that one can add a significant fragment
  of \(M_1\)'s iteration strategy to \(M_1\) without destroying the Woodinness of \(\delta\). How much exactly can be added?

• Octover 15th - Farmer Schlutzenberg, Strategy extensions of \(M_1\)

  Continuation

• October 22nd - Ralf Schindler. A universe where \(NS_{\omega_1}\) is saturated, \(\Delta_1(\omega_1)\)-definable and \(MA_{\omega_1}\) holds.

  Abstract: The definability of the nonstationary ideal can serve as a test question for the 'good' behaviour of subsets of \(\omega_1\). Recent results of Larson, Schindler and Wu and Aspero-Schindler indicate that strong forcing axioms imply that \(NS_{\omega_1}\) behaves well, i.e. can not be \(\Delta_1(\omega_1)\)-definable. We show that on the other hand Martin’s Axiom is consistent with the non stationary ideal being \(\Delta_1(\omega_1)\)-definable and saturated, assuming the existence of a Woodin cardinal. This work builds heavily on a coding technique which forces first a suitable ground model and in a second step does the coding over the generically created universe. Goal of the talk will be to highlight some of its properties.

• October 29th - Ralf Schindler. \(NS_{\omega_1}\) is not \(\Pi_1\)

• November 5th - Ralf Schindler. \(NS_{\omega_1}\) is not \(\Pi_1\)

  Continuation

• November 12th - Ralf Schindler. \(NS_{\omega_1}\) is not \(\Pi_1\)

  Continuation

• November 26th - Ralf Schindler. \(NS_{\omega_1}\) is not \(\Pi_1\)

  Continuation

• December 3rd - Rahman Mohammadpour (Paris). Some Combinatorial Properties Concerning Guessing Models

  Abstract: C. Weiß formulated some combinatorial properties that capture the essence of some large cardinal properties, mostly at level of supercompactness, but can hold at small cardinals. He and M. Viale could then use them to show that any standard forcing construction of a model of PFA requires at least a strongly compact cardinal. In this talk, I will introduce \(GM+(\omega_3,\omega_1)\) a combinatorial principle strengthening those by Weiß, and will discuss also the consequences of it. This is joint work with B. Velickovic.

• December 13th - Vladimir Kanovei (Moscow). What can hold exactly on the \(n\)th projective level?

• December 17th - Liuzhen Wu (Beijing).

• January 14th - Liuzhen Wu (Beijing),

   Continued

• January 21st - Farmer Schlutzenberg. Reinhardt embeddings and non-definability

  Abstract: Reinhardt embeddings and non-definability: A Reinhardt cardinal is the
  critical point of a Reinhardt embedding, that is, a non-trivial
  elementary embedding \(j:V\rightarrow V\). Kunen showed that there is no Reinhardt
  embedding - assuming ZFC. They are, however, not known to be
  inconsistent with ZF alone - at least when formulated correctly. In
  the last few years, Woodin, Koellner, Bagaria and Cutolo have
  investigated Reinhardt and stronger larger cardinals (all incompatible
  with AC). We will discuss these large cardinal principles, and prove
  some non-definability consequences of them, including that if there is
  a Reinhardt cardinal then \(V\) is not of the form \(HOD(X)\) for a set X. We
  will also discuss the question as to whether V_delta can support a
  "local" Reinhardt embedding \(j:V_\delta\rightarrow V_\delta\) with \(j\) in \(L(V_\delta)\)
  -- while I do not know the answer to this question, I will show that
  such a \(j\) cannot be constructed extremely soon above \(V_\delta\).

• January 31st - Assaf Rinot (Bar-Ilan University)

Talks during the summer of 2019

•  April 9th - Farmer Schlutzenberg. The core model in mice

  Abstract: We will discuss a proof showing that if \(M\) is a mouse below a superstrong which has a least Woodin cardinal \(\delta_0\) and \(M|\delta_0\) satisfies ``I am fully iterable'', then \(M|\delta_0\) satsifies ``\(V=K\)''. As a corollary, (i) for any \(M\)-generic \(G\) for a poset \(P\in V_{\delta_0}^M\), we have \(M|\delta_0 \subseteq HOD^{M[G]}\), and (ii) \(M\) has no proper ground via a forcing in \(V_{\delta_0}^M\).
  *These results may already be known to others, but I am not aware of a written account up until now.

•  April 16th - Farmer Schlutzenberg. The core model in mice

  Continuation

• April 23rd - Ralf Schindler. \(\mathrm{MM}^{++} \implies (*)\)

  where \((*)\) is the statement: \(\mathrm{AD}^{L(\mathbb{R})}\) holds and there is some \(g \in V\) which is \( \mathbb{P}_{\mathrm{max}}\)-generic over \(L(\mathbb{R})\) such that \(H_{\omega_2}^V \subseteq L(\mathbb{R})[g]\).

• May 3rd - Filippo Calderoni. The complexity of the bi-embeddability relation beyond Borel reductions

  Update: We meet 16:00 in lecutre hall M4.

  Abstract: In this talk we shall investigate the bi-embeddability relation on countable periodic groups with techniques from (descriptive) set theory. First we shall show that bi-embeddability is not a Borel equivalence relation in the case of primary groups. Next we shall use forcing and the theory of pinned names to show that the isomorphism and bi-embeddability relations on countable periodic groups are incomparable up to Borel reducibility. To contrast this result, we shall discuss how bi-embeddability is strictly simpler than isomorphism under \(\Delta^1_2\) reducibility and some mild large cardinal assumptions. This is joint work with Simon Thomas.

• May 8th - Dima Sinapova (UIC). Singularizing cardinals

  UPDATE: WE MEET 16:00 IN LECUTRE HALL M5.
  
  Abstract: It is well known that if \(\kappa\) is inaccessible in \(V\) and \(W\) is an outer model of \(V\) such that \((\kappa^+)^V = (\kappa^+)^W\) and \(\mathrm{cf}^W(\kappa) = \omega\), then \(\square_{\kappa,\omega}\) holds in \(W\). Many strengthenings of this theorem have been obtained as well. We prove that this theorem does not generalize to uncountable cofinalities. Using Magidor's forcing, we show that we can singularize \(\kappa\) in a cardinal preserving way, and have that in the outer model \(\square_{\kappa,\tau}\) fails for all \(\tau<\kappa\).
• This is joint work with Maxwell Levine

• May 14th - Ralf Schindler. \(\mathrm{MM}^{++} \implies (*)\)

  Continuation

• May 21st - Ralf Schindler. \(\mathrm{MM}^{++} \implies (*)\)

  Continuation

Talks during the winter of 2018

• August 28th - Liuzhen Wu (Chinese Academy of Sciences). Lifting argument for Neeman's forcing with side condition

  Abstract: Neeman introduces the generalized side condition forcing using two type of models. In this talk, we will describe a lifting argument for a modified version of Neeman's forcing in presence of huge cardinals. We will then discuss the similarity between Neeman's forcing and Kunen-style forcing construction in the realm of hugeness. 

• October 9th - Dominik Adolf. Determinacy from threadability at and below \(\Theta\).

  Abstract: In this talk we consider the hypothesis that all cardinals less or equal to \(\Theta\) are threadable. (As usual \(\Theta\) refers to the supremum of prewellorders on the set of reals.) The hypothesis follows easily from \(\mathrm{AD}_\mathbb{R}\). We will present a way to perform the "next Woodin" step of the core model induction from the hypothesis under some light extra assumptions. We feel confident that the proof can be extended into an equiconsistency.

• October 26th - Gabriel Fernandes. Local core models with more Woodin cardinals without the measurable

  Abstract: Assuming \(\kappa\) is a singular cardinal such that \(\kappa = \beth_{\kappa}\) and that there is no premouse \(\mathcal{M}\) such that \(\mathcal{M} \cap \mathrm{OR} = \kappa\) and \(\mathcal{M} \models \forall \alpha \exists \delta > \alpha \colon \delta \text{ is a Woodin cardinal}\), we isolate the core model up to \(\kappa\). If time permits we will mention a few applications. These reults are part of my Ph.D. thesis.

• October 30th - Stefan Hoffelner. Coding, Forcing Axioms and the Definability of the non stationary Ideal

  Abstract: We will introduce and discuss a coding technique which is a combination of a result of Shelah and David’s trick. This method can be used in the context of inner models which have a certain amount of generic absoluteness and works for arbitrarily long iterations. We plan to sketch some applications.

• November 6th - Stefan Hoffelner. Coding, Forcing Axioms and the Definability of the non stationary Ideal

  Continuation

• November 13th - Ralf Schindler. Hamel bases without a wellorder of the reals

• November 20th - Farmer Schlutzenberg. Grounds of mice via \(\sigma\)-closed forcings

  Abstract: Abstract: If \(W \models\mathrm{ZFC} \) and \(W\) is a non-trivial ground of a mouse \(M\) via a forcing \(\mathbb{P} \in W\) which is \(\sigma\)-closed in \(W\) then \(M|\omega_1^M\notin W\).

• November 27th - Farmer Schlutzenberg. Grounds of mice via \(\sigma\)-closed forcings

  Continuation

• December 4th - Farmer Schlutzenberg. Grounds of mice via \(\sigma\)-closed forcings

  Continuation

• December 11th - Ralf Schindler. Jensen forcing and the Sacks property

  Abstract: Jensen introduced a variant of Sacks forcing which adds exactly one generic real. We introduce a variant of Jensen's forcing which also has the Sacks property. 

• January 11th - Joel David Hamkins, University of Oxford. An infinitary-logic-free proof of the Barwise end-extension theorem, with new applications
  
  Update: We will meet at 14:15 in lecture hall M5.

  Abstract: Abstract. I shall present a new proof, with new applications, of the amazing extension theorem of Barwise (1971), which shows that every countable model of \(\mathrm{ZF}\) has an end-extension to a model of \(\mathrm{ZFC}+V=L\).  This theorem is both (i) a technical culmination of Barwise's pioneering methods in admissible set theory and the admissible cover, but also (ii) one of those rare mathematical results saturated with significance for the philosophy of set theory.  My new proof uses only classical methods of descriptive set theory, and makes no mention of infinitary logic. The results are directly connected with recent advances on the universal \(\Sigma_1\)-definable finite set, a set-theoretic version of Woodin's universal algorithm. 

• January 22th - Andreas Lietz. The \(<\kappa\)-mantle.

  Abstract: Abstract: The \(<\kappa\) mantle is a restricted version of the usual geologic mantle. Instead of intersecting all grounds, only those which extend to \(V\) via \(\kappa\)-small forcing are considered. We investigate at which cardinals \(\kappa\) the \(<\kappa\) mantle is a model of \(\mathrm{ZFC}\). This is joint work with Ralf Schindler.

Talks during the summer of 2018

• April 10th - Andreas Lietz. Set-Theoretic Geology

  Abstract: Set-Theoretic Geology is the study of grounds, the ground models of forcing extensions, and the generic multiverse and was initially founded by Hamkins and Reitz in an effort to find regular structure under the generic "dust" added by forcing. Although their hope was not quite fulfilled, this investigation left many interesting questions about the nature of forcing open until the recent results of Usuba about the strong downwards directed grounds hypothesis. For example the mantle, the intersection of all grounds, turned out to be a model of ZFC and the largest forcing invariant definable class. I want to give an exposition of this subject and discuss the interplay between the generic multiverse, its mandtle and large cardinals, including Usubas hyper-huge cardinals.

• April 17th - Stefan Mesken. Varsovian Models with more Woodin cardinals part I

  Abstract: Let \(M\) be the least pure extender model with a strong cardinal that itself is a limit of Woodin cardinals. We calculate the mantle of \(M\) and show that it is in fact a fine structural bedrock.

• April 24th - Stefan Mesken. Varsovian Models with more Woodin cardinals part II

• May 5th - Stefan Mesken. Varsovian Models with more Woodin cardinals part III

• May 29th - Stefan Hoffelner: t.b.a.

• June 5th - Jinglun Cai: On the consistency bound of a square partition instance

  Abstract: We show that certain square partition instance follows from a second order Löwenheim-Skolem statement, which is in turn implied by the existence of a huge cardinal. Hence hugeness is found to be a better consistency bound of the square partition instance.

• June 12th - Stefan Hoffelner: A model for \(\mathrm{NS}_{\omega_{1}}\) saturated, \(\Delta_{1}\)-definable and Martin's Axiom

  Abstract: I will present a sketch of a new coding method which can be used over an arbitrary ground model and will result in a model where the nonstationary ideal on \(\omega_{1}\) will become \(\Sigma_{1} (C)\)-definable,  where \(C\) is a ladder system on \(\omega_{1}\). When applied in the context of universes of "\(\mathrm{NS}_{\omega_{1}}\) is saturated" it is possible, under the assumption of stationarily many Woodin cardinals, to build a model where additionally Martin's Axiom will hold.

• June 19th - Farmer Schlutzenberg: Extending Iteration Strategies

  Abstract: Let \(\kappa\) be regular uncountable and \(\Sigma\) be an \((n,\kappa+1)\)-strategy (for normal trees) for an \(n\)-sound premouse
  \(M\). If \(\Sigma\) has a certain natural condensation property then \(\Sigma\) can be extended to an \((n,\kappa,\kappa+1)\)-strategy (for
  stacks of normal trees), and can also be extended to an \((n,\kappa+1)\)-strategy \(\Sigma'\) of \(V[G]\), whenever \(G\) is \(V\)-generic
  for a \(\kappa\)-cc forcing. The condensation property follows from the (weak) Dodd-Jensen property. We will discuss such condensation and how
  the extensions of \(\Sigma\) are constructed.

• June 26th - William Chan (Univ. Northern Texas): The Destruction of the Axiom of Determinacy by Forcing on \(\mathbb{R}\) when \(\Theta\) is Regular.

  Abstract: Ikegami and Trang have shown that many known forcings, such as Cohen forcing, can not preserve \(\mathrm{AD}\). They showed that in natural models of determinacy satisfying \(\mathrm{ZF} + \mathrm{AD}^+ + V = L(\mathcal{P}(\mathbb{R}))\), forcings that preserve \(\mathrm{AD}\) must preserve \(\Theta\). However, they also showed it is consistent (with some additional assumptions) that a forcing can preserve \(\mathrm{AD}\) and increase \(\Theta\), which implies a new set of reals has been added.
  We will show, assuming \(\mathrm{ZF}\) and \(\mathrm{AD}\), that every nontrivial wellorderable forcing of cardinality less than \(\Theta\) forces the failure of \(\mathrm{AD}\). Assuming \(\mathrm{ZF}\), \(\mathrm{AD}\), and \(\Theta\) is regular, we will show that every nontrivial forcing which is a surjective image of the reals forces the failure of \(\mathrm{AD}\). This is joint work with Stephen Jackson.

• July 3rd - William Chan (Univ. Northern Texas): Suslin lines under \(\mathrm{AD}\)

  Abstract: A Suslin line is a nonseparable complete dense linear ordering without endpoints which has the countable chain condition. We will show that \(\mathrm{ZF} + \mathrm{AD}^+ + V = L(\mathcal{P}(\mathbb{R}))\) proves there are no Suslin lines. In particular, if \(L(\mathbb{R})\) is a model of \(\mathrm{AD}\), then \(L(\mathbb{R})\) has no Suslin lines, which answers a question of Foreman. This is joint work with Stephen Jackson.

Talks during the winter of 2017

• October 16th  - Farmer Schlutzenberg. \(\mathrm{HOD}^{M[g]}\).

  Abstract: Let \(M = M_{SWSW}\) be the least pure extender model with \(\delta_0^M < \kappa_0^M < \delta_1^M < \kappa_1^M\), where \(\delta_i^M\) is a Woodin cardinal and \(\kappa_i^M\) is a strong cardinal in \(M\). We aim to analyze \(\mathrm{HOD}^{M[g]}\), where \(g\) is a generic for \(\mathrm{Coll}(\omega, < \kappa_0^M)\), and show that in fact \(\mathrm{HOD}^{M[g]} = \mathcal{V_0^M} = \mathcal{M}_{\infty}[*]\) - the \(0\)-order Varsovian model of \(M\).

• October 23th  - Farmer Schlutzenberg. \(\mathrm{HOD}^{M[g]}\) continued.

• November 7th - Yizheng Zhu. The internal structure of \(\mathrm{HOD}^{L[x]}\) up to its Woodin [retracted]

  Abstract: Assume \(\boldsymbol{\Delta}^1_3\)-determinacy. It is shown that for any \(x \geq_T M_1^{\#}\), \(\mathrm{HOD}^{L[x]}\) is a model of GCH, and in fact, it is a Jensen-Steel core model up to \(\omega_2^{L[x]}\).

• November 28th - Dominik Adolf. Structures with alternating cofinalities - some lower bounds.

• December 5th - Farmer Schlutzenberg. \(\mathrm{HOD}^{M[g]}\) continued.

• December 12th - Farmer Schlutzenberg. \(\mathrm{HOD}^{M[g]}\) continued.

• December 19th - Ralf Schindler. The possible number of Woodin cardinals in core models.

  Abstract: There are only three such numbers: 0, 1, and strongly many. This is joint work with Grigor Sargsyan.

• January 9th - Aleksandra Kwiatkowska. Ramsey theory, topological dynamics and Ważewski dendrites

  Abstract: We study homeomorphism groups of compact connected spaces called Ważewski dendrites, focusing on their universal minimal flows. For each \( P \subset \{3,4,...,\omega\} \) there is a unique up to homeomorphism Ważewski dendrite \(W_P\). If \(P\) is finite, we prove that the universal minimal flow of \(\mathrm{Hom}(W_P)\) is metrizable and we compute it explicitly. This answers a question of B. Duchesne. If \(P\) is infinite, we show that the universal minimal flow of \(\mathrm{Hom}(W_P)\) is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a non-metrizable universal minimal flow.

• January 16th - Grzegorz Plebanek. Mardešić problem and free productsof Boolean algebras

  Abstract: Years ago Mardešić posed a certain problem on continuous images of linearly ordered compact spaces.  In a particular case, it amounts to asking what kind of free products of Boolean algebras can be embedded into free products of interval algebras.
  In a recent paper written with Gonzalo Martínez Cervantes we introduced a new kind of dimension of compacta, combinatorial in nature. This concept enabled us to answer Mardešić's question and present  generalizations of some results on Boolean algebras due to Heindorf.

• January 23rd - Stefan Hoffelner.  A model where NS is saturated and $\(\Delta_1\)$-definable

  Abstract: Questions which investigate the interplay of the saturation of the nonstationary ideal on \(\omega_1\), NS, and definability properties of the surrounding universe can yield surprising and deep results. Woodins theorem that in a model with a measurable cardinal where NS is saturated, CH must definably fail is the paradigmatic example. It is another remarkable theorem of H. Woodin that given \(\omega\)-many Woodin cardinals there is a model in which NS is saturated and \(\omega\)-dense, which in particular implies that NS is (boldface) \(\Delta_1\)-definable. The latter statement is of considerable interest in the emerging field of generalized descriptive set theory, as the club filter is known to violate the Baire property. With that being said the following question, asked first by S.D. Friedman and L. Wu seems relevant: Is it possible to construct a model in which NS is both \(\Delta_1\)-definable and saturated from less than \(\omega\)-many Woodins? In this talk I will outline a proof that this is indeed the case: Given the existence of \(M_1^{\#}\), there is a model of ZFC in which the nonstationary ideal on \(\omega_1\) is saturated and \(\Delta_1\)-definable with parameter \(\omega_1\). In the course of the proof I will present a new coding technique which seems to be quite suitable to obtain definability results in the presence of iterated forcing constructions over inner models for large cardinals. See arXiv:1701.07230