Schedules and Abstracts – Week 1 at ESI

Schedule for the first week (at ESI)

Abstracts for the first week (at ESI)

John Steel: The comparison lemma

We shall discuss the comparison lemma of inner model theory, at a level accessible to non-experts.

Jouko Väänänen: Inner models constructed from generalized logics and their relationship with the standard inner models

This talk is the first of a series of two talks. The second talk will be delivered by Menachem Magidor.
For any extension L* of first order logic, e.g. by generalized quantifiers, we associate an inner model C(L*) obtained as the inner model L of constructible sets but using the logic L* in place of first order logic. For many but not all logics L* the inner model C(L*) is simply L. Interesting examples are the generalized quantifier which says that a definable linear order has countable cofinality giving rise to an inner model we denote by C*. Another interesting example is the second order quantifier which says that a formula is satisfied by a club of countable subsets of the domain of the model, giving rise to an inner model we call C(aa). Clearly C* is a submodel of C(aa). Assuming a proper class of Woodin cardinals, C* has a proper class of weakly compact cardinals and its theory is forcing absolute. Under the same assumption, C(aa) has a proper class of measurable cardinals, it satisfies GCH and its theory is again forcing absolute. In this two part talk we compare the models C* and C(aa) as well as some of their cousins to the extender models of inner model theory and their mice. It turns out that our inner models have a closer relationship with extender models and their mice than meets the eye. In many cases one can determine a particular mouse that projects to omega which is associated with a given C(L*). It is the minimal mouse (in the mice order) which is not in C(L*). It can be considered as a gauge for the strength of the logic L*. In these two talks we shall survey some of the ideas which are used in the analysis of inner models of the form C(L*) and their relations to the usual fine structural models.

Bartosz Wcislo: Separating levels of DC on reals from levels of PD.

In a recent article by Friedman, Gitman, and Kanovei, it was shown that AC does not imply DC in the context of second-order arithmetic. The proof was based on a constrution of a certain symmetric extension of L, using Jensen’s Diamond Principle. In our talk, we will show that by modifying their construction, we can obtain a model of ZF in which, for a given n, Pi^1_n-determinacy holds, but Pi^1_k dependent choice for real numbers fails for some explicitly given value of k.
This presentation is based on joint work with Sandra Müller.

William Chan: Cardinality of the Set of Bounded Subsets of a Cardinal

This talk will discuss the cardinality of the set of bounded subsets of a cardinal below Theta under Woodin’s theory AD+. This will be applied to study the cardinality of cardinal exponentiation below Theta in the theory AD+.

Menachem Magidor / Ralf Schindler: Inner Models constructed from generalized logics and thier relationship wiht the standard models.

This talk is the first of a series of two talks. The first talk will be delivered by Jouko Väänänen.
For any extension L* of first order logic, e.g. by generalized quantifiers, we associate an inner model C(L*) obtained as the inner model L of constructible sets but using the logic L* in place of first order logic. For many but not all logics L* the inner model C(L*) is simply L. Interesting examples are the generalized quantifier which says that a definable linear order has countable cofinality giving rise to an inner model we denote by C*. Another interesting example is the second order quantifier which says that a formula is satisfied by a club of countable subsets of the domain of the model, giving rise to an inner model we call C(aa). Clearly C* is a submodel of C(aa). Assuming a proper class of Woodin cardinals, C* has a proper class of weakly compact cardinals and its theory is forcing absolute. Under the same assumption, C(aa) has a proper class of measurable cardinals, it satisfies GCH and its theory is again forcing absolute. In this two part talk we compare the models C* and C(aa) as well as some of their cousins to the extender models of inner model theory and their mice. It turns out that our inner models have a closer relationship with extender models and their mice than meets the eye. In many cases one can determine a particular mouse that projects to omega which is associated with a given C(L*). It is the minimal mouse (in the mice order) which is not in C(L*). It can be considered as a gauge for the strength of the logic L*. In these two talks we shall survey some of the ideas which are used in the analysis of inner models of the form C(L*) and their relations to the usual fine structural models.

Dima Sinapova: Stationary reflection for $\aleph_{\omega_1+1}$

In recent years there has been a lot of progress in combining stationary reflection at the successor of a singular and the failure of SCH. The motivation goes back to two classical results of Magidor: tationary reflection at $\aleph_{\omega+1}$ can be force from large cardinals; and from large cardinals, one can get the failure of SCH at $\aleph_\omega$. In this talk we will focus on the case when the singular cardinal has uncountable cofinality. We will show that from supercompact cardinals, there is a Prikry type iteration, such that in the final model we have stationary reflection at $\aleph_{\omega+1}$ together with the failure of SCH at $\aleph_{\omega_1}$. This is joint work with Tom Benhamou.

Benjamin Siskind: Order-preserving Martin’s Conjecture and Inner Model Theory

Martin’s Conjecture is a proposed classification of Turing-invariant functions under the Axiom of Determinacy. Whether the classification holds for the ostensibly smaller class of order-preserving functions is open, but more tractable. For example, it is known that the conjecture holds restricted to the Borel order-preserving functions. In this talk, we’ll explain an approach to proving Martin’s Conjecture for order-preserving functions beyond the Borel ones via inner model theory. This is joint work with Patrick Lutz.

Martin Zeman: On the failure of two successive squares

Given a cardinal \kappa we consider the statement “\kappa is threadable + \square_\kappa fails”. Assuming “CH + \kappa=\omega_2” or “2^\omega=\omega_2 + \kappa=\omega_3”, a core model induction argument then yields “AD_R + \Theta is regular”. This brings the results by Jensen-Schindler-Schimmerling-Steel one cardinal down. This is a joint work with Nam Trang.

Dominik Adolf: Chang’s Conjecture and Mouse Reflection

The so-called Strong Chang’s Conjecture $(\aleph_3,\aleph_2) \twoheadrightarrow (\aleph_2,\aleph_1)$ is generally believed to be a very strong property possibly approaching the strength of a huge cardinal. Unfortunately, inner model theory so far has failed in corrobariting this intuition. In the first part of this talk we will give a quick overview of the state of the art and furthermore try to elucidate the problems that have blocked progress so far.
In the second part we will introduce what we call the Long Chang’s Conjecture which is simply a Chang’s conjecture involving infinitely many cardinals. We will then explain how this Long Chang’s Conjecture allows us to side-step aforementioned problems and achieve Projective Determinacy. Of particular interest in this argument is the way we achieve mouse reflection in the context of the core model induction. An approach we believe will also be useful when considering the more traditional forms of Chang conjectures.

Andreas Lietz: On Mathias Characterizations for Generics for Variants Of Namba Forcing

There are many different variations of Namba forcing, for example the standard formulation Nm consisting of $\omega_2$-perfect trees and the variation Nm’ of those $\omega_2$-perfect tree which split everywhere above their stem. Magidor-Shelah have shown, assuming CH, that Nm is essentially different from Nm’ Jensen proved an even stronger theorem in which he differentiates Nm and Nm’ further from the variant of Nm’ consisting of those trees in Nm’ all of whose nodes above the stem have stationarily many immediate successors. Jensen also assumed CH.

We generalize these theorems by removing the CH assumption and taking into account many more variations of Namba forcing. Further, we show that all “natural” variations of Namba forcing generate extensions which are minimal conditioned on $\cof(\omega_2^V)=\omega$ and moreover, we analyze exactly which and how many other sequences in such an extension are generic for a variation of Namba forcing. Further, we show that no Mathias-style characterization for variants of Namba forcing are possible except for Priky-style forcings. This answers a question of Gunter Fuchs.

Takehiko Gappo: Determinacy of long games just beyond fixed countable length

Martin and Harrington showed that the analytic determinacy of games on the natural numbers of length $\omega$ is equivalent to the existence of $x^{\sharp}$ for reals x.
The generalization of this for longer games is also known: Neeman and Trang–Woodin’s results say that for any $2\leq\alpha<\omega_1$, the analytic determinacy of games of length $\omega^{\alpha}$ is equivalent to the sharp for an inner model with $\omega^{-1+\alpha}$ many Woodin cardinals.
We show that such equivalence also holds for games of variable countable length, in which the length of a play is still countable but determined by the moves.
A game introduced in the talk can be viewed as the “diagonalization” of all games of fixed countable length and its analytic determinacy turns out to be equivalent to the sharp for an inner model with ?$\lambda$ many Woodin cardinals, where ?$\lambda$ is the order type of Woodin cardinals below $\lambda$?.
This is an ongoing joint work with Sandra Müller.

Trevor Wilson: Characterizing large cardinals in terms of Löwenheim–Skolem and weak compactness properties of strong logics

Hugh Woodin: Exotic Models

Juan Santiago Suarez: Forcing and consistency properties

The relation between forcing, Boolean valued models and consistency properties was implicit from the beginning of forcing in the works of Mansfield, Keisler, Solovay, Scott and Makkai. However, no concrete application was given. The aim of this talk is to argue that consistency properties and infinintary logics provide a natural setting for building forcing notions. First, we will present general logic results such as completeness, interpolation and omittying types. All can be proved for the most general case, the logic $\mathcal{L}_{\infty \infty}$. Nonetheless, wether or not these theorems can be proved with respect to Boolean valued models with the mixing property allows to strictly separate $\mathcal{L}_{\infty \omega}$ from $\mathcal{L}_{\infty \infty}$. Second, we will analyse the relation between consistency properties and forcing. Motivated by the fact that every generic filter can be seen as the model built by a consistency property, we will discuss for what formulas there is a consistency property forcing a model in a nice way. The results presented here build on the proof of MM$^{++}$ implies $(*)$ by Asperò and Schindler together with the AS condition recently isolated by Kasum and Velickovic.

Jan Kruschewski: Analysis of HOD for Admissible Structures

Let $n \geq 1$ and assume that there is a Woodin cardinal. For $x \in \R$ let $\alpha_x$ be the least $\beta$ such that
\[  L_\beta [x] \models \Sigma_n \text{-} \kp + \exists \kappa (“\kappa \text{ is inaccessible and }\kappa^+ \text{ exists}”).  \]
Then there is a cone of reals $x$ such that letting $\kappa$ be the inaccessible of $L_{\alpha_x}[x]$, $G \subset \col(\omega, <\kappa)$ be $(L_{\alpha_x}[x],\col(\omega, <\kappa))$-generic, $\Sigma_n$-$\hod$ be the class of elements $a$ of $L_{\alpha_x}[x,G]$ such that $\tc(\{a\}) \subset \Sigma_n\text{-}\od$, where $\Sigma_n\text{-}\od$ is the class of all elements which are ordinal definable over $L_{\alpha_x}[x,G]$ via a $\Sigma_n$-formula, we have that $\Sigma_n$-$\hod$ is an iterate of a mouse adjoined with a fragment of its iteration strategy. Moreover, $\Sigma_n$-$\hod \models \Sigma_n \text{-} \kp$, $\omega_2^{L_{\alpha_x}[x,G]}$ is Woodin in $\Sigma_n$-$\hod$ and $\Sigma_n$-$\hod$ is a forcing ground of $L_{\alpha_x}[x,G]$.

Derek Levinson: Unreachability in the Second Projective-Like Hierarchy

For $\Gamma = \Sigma_{2n+3}(J_2(\mathbb{R}))$, we show there is no sequence of distinct sets in $\Gamma$ of length $\delta_\Gamma$. Our proof builds on the analysis of the mice $M_n^{ld}$ performed by Rudominer, Steel, and Woodin. This is joint work with Itay Neeman.

James Cummings: Squares, scales and lines

Todorcevic showed that if kappa is singular of cofinality omega and square_kappa holds then there is a linear order of cardinality kappa^+ which is not sigma-scattered while all of its small suborders are sigma-wellordered. I will discuss how to get the same result from some pcf-theoretic principles, which follow both from weaker forms of square and from certain failures of SCH. If time allows I will discuss some related problems.

Grigor Sargsyan: Hod mice as a bridge between determinacy, forcing axioms and infintary combinatorics

In this talk we will outline recent developments in descritptive inner model theory that have applications in the theory of forcing axioms, in determinacy theory and in infintary combinatorics. In particular, we will concentrate on the following four themes.
1. Cofinality of Theta^{L(uB)} in models of Forcing Axioms and in models of Sealing. We will show that Sealing doesn’t decide thie value of this cofinality. In particular, letting kappa be this cofinality, we will show that T_i=Sealing+kappa=omega_i is consistent for i=1, 2, and 3. This is joint work with Douglas Blue and Matteo Viale.
2. We will show that omega_1 is <Theta-Berkeley in models of AD^+. This is a joint work with Douglas Blue.
3. For each n< omega, we will force the theory ZFC+MM^{++}(c)+for all i\leq n+2( failure of square_{omega_i} + failure of square(omega_i)) over a model of determinacy. This is a joint work with Paul Larson and Douglas Blue.
4. We will outline some basic theory of Nairian Models that suggest the possibility of forcing even stronger ideals on omega_1.
Clause 3 above has a consequence that the K^c constructions with 2^omega closed background certificates can consistently fail to converge.

Gabriel Goldberg: The HOD conjecture and its failure

The subject of this tutorial is Woodin’s HOD conjecture, one of the most prominent open problems in pure set theory. We begin with a proof of his HOD dichotomy theorem along with an improvement of the speaker’s reducing the large cardinal hypothesis from an extendible to a strongly compact cardinal. Following this, we mostly discuss the implications of the failure of the HOD conjecture, especially $\omega$-strongly measurable cardinals and a condition under which such cardinals are locally supercompact in $\HOD$.

Taichi Yasuda: Martin’s Maximum^{*, ++}_{c} in P_{max} extensions of strong models of determinacy

We study a strengthening of MM^{++} which is called MM^{*, ++} and which was introduced by Asperó and Schindler. We force its bounded version MM^{*, ++}_{c}, which is stronger that both MM^{++}(c) as well as BMM^{++}, by P_{max} forcing. We also give the construction of the ground model, which builds upon Gappo and Sargsyan, and the derived model construction of Larson, Sargsyan, and Wilson. This is a joint work with Ralf Schindler.

Farmer Schlutzenberg: Ladder mice

We will discuss a new analysis of ladder mice, first introduced and studied by Rudominer, and then Woodin and Steel. Our analysis establishes a (lightface) mouse set theorem, which appears to be more general than what was known earlier: OD_{alpha n} is a mouse set for every ordinal alpha of countable L(R)-cofinality such that [alpha,alpha] is a projective-like gap and alpha is not the successor of a strong gap, and for every integer n>=1. The analysis also gives an alternate proof of this in the case “just past projective”, avoiding the stationary tower. It also establishes an associated on-a-cone “anti-correctness” result. Anti-correctness is the generalization of, for example, the facts that (Pi^1_3)^V truth about reals in M_1 is (Sigma^1_3)^{M_1}, and that (Pi^1_3)^{M_1} truth (about reals in M_1) is (Sigma^1_3)^V. Time permitting, we may also mention a version of ladder mice at the end of a weak gap / successor of a strong gap. This work appears to be a useful component toward a positive resolution of the Rudominer-Steel conjecture on optimal wellorders of the reals, a related question on which the author and Steel are working.

Monroe Eskew: Transferring ideals

In recent work with Hayut, we produced a model where for all n > 0, there is a normal ideal $I$ on $\omega_n$ with $P(\omega_n)/I$ forcing equivalent to $Col(\omega_{n-1},\omega_n)$. The most interesting consequences of this stem from a transfer theorem of Woodin that allows us to find copies of boolean algebas of the form $P(\omega_n)/I$ in ones of the form $P(\omega_m)/J$ for $m>n$ and with $J$ uniform and having the same additivity as $I$. We will sketch the proof of Woodin’s theorem, raise the question of whether some these transfers can be witnessed by certain surjections that give rise to a direct limit generic ultrapower, and discuss what this could mean for further combinatorial applications.

Nam Trang: Almost disjoint families in natural models of AD^+

For each cardinal κ, let B(κ) be the ideal of bounded subsets of κ and P_κ(κ) be the ideal of subsets of κ of cardinality less than κ. Assuming AD^+, for all κ < Θ, there are no maximal B(κ) almost disjoint families A such that ¬(|A| < cof(κ)). For all κ < Θ, if cof(κ) > ω, then there are no maximal P_κ(κ) almost disjoint families A so that ¬(|A| < cof(κ)).

This is joint work with W. Chan and S. Jackson. Our work is inspired by work of Schrittesser and T¨ornquist, and of Neeman and Norwood that showed there are no maximal almost disjoint families on ω under AD^+.

Shervin Sorouri: Prikry sequences of longer length

We shall show that if E is a nice enough extender and we iterate V via E and it’s images countably many times, and if A is any countable set of generators of E, then the thread of A is generic over the final model in a cardinal preserving way. This is a generalization of the usual prikry forcing base on ideas of Itay Neeman, and at the same time it is a refinement of a general result of Itay Neeman in the case of linear iterations.