© Gregor Schindler 2019

Berkeley conference on

inner model theory

July 08--19, 2019

Organizers: Ralf Schindler (Münster) and John Steel (Berkeley).

This conference will be a sequel to the 1st Conference on the core model induction and hod mice that was held in Münster (FRG), July 19 -- August 06, 2010, to the 2nd Conference on the core model induction and hod mice that was held in Münster (FRG), August 08 -- 19, 2011, to the AIM Workshop on Descriptive Inner Model Theory, held in Palo Alto (CA), June 02 -- 06, 2014, to the Conference on Descriptive Inner Model Theory, held in Berkeley (CA) June 09 -- 13, 2014 to the 3rd Münster Conference on inner model theory, the core model induction, and hod mice that was held in Münster (FRG), July 20 -- 31, 2015, to the 1st Irvine conference on descriptive inner model theory and hod mice that was held in Irvine (CA), July 18 -- 29, 2016, to the 4th Münster Conference on inner model theory, the core model induction, and hod mice that was held in Münster (FRG), July 17 -- August 01, 2017, as well as to the 1st Girona conference on inner model theory that was held in Girona (Catalonia), July 16 -- 27, 2018.

Once more, this conference will now draw together researchers and advanced students with an interest in inner model theory, in order to communicate and further explore recent work. There will be courses and single talks.
We will meet Monday--Friday, with 2 1/2 hours of lectures in the morning and 2 1/2 hours of lectures in the afternoon. This will leave ample time for problem sessions, informal seminars, and other interactions.

We gratefully acknowledge financial support through an NSF grant, by the Berkeley Group in Logic and the Methodology of Science, and through a DFG grant (Mathematics Münster: Dynamics - Geometry - Structure).

The talks will take place 748 Evans Hall, and breaks will be in the Alfred Tarski room, 737 Evans Hall.


Mon, July 08 Tue, July 09Wed, July 10Thu, July 11 Fri, July 12
9:30--10:45 Schindler: part 1 Schindler: part 3Steel: part 1Zeman: part 1Schimmerling: part 1
11:15--12:30 Schindler: part 2 Schindler: part 4Steel: part 2Zeman: part 2 Schimmerling: part 2
14:30--15:45 Goldberg Ben-Neria 1Adolf: part 1GitikRudominer: part 1
16:15--17:30 Jackson Ben-Neria 2Adolf: part 2SiskindRudominer: part 2
17:30--∞ Problems and Discussions Problems and Discussions Problems and Discussions Problems and Discussionsfree

Mon, July 15 Tue, July 16Wed, July 17Thu, July 18 Fri, July 19
9:30--10:45 Trang: part 1 Chan: part 1Schlutzenberg: part 3Shi---
11:15--12:30 Trang: part 2 Chan: part 2Schimmerling: part 3Wilson II---
14:30--15:45 Wilson I Schlutzenberg: part 1Trang: part 3Fernandes---
16:15--17:30 --- Schlutzenberg: part 2Steel: part 3Trang: part 4---
17:30--∞ Problems and Discussions Problems and DiscussionsProblems and DiskussionsProblem Sessionfree

Problem list, talks, and abstracts:

  • List of open problems.

  • Dominik Adolf (2 slots): LSA from failure of covering at singular strong limits. Abstract: Steel has shown that the failure of square at a singular strong limit implies that the axiom of determinacy holds in L(R). Sargsyan later strengthened this by showing that there must exist a model of determinacy containing all the reals and a prewellorder on the reals that is not ordinal definable from a real parameter. We shall continue this work: utilizing recent work by Sargsyan and Trang we will show that there exist models of determinacy in which θ (, the supremum of the length of pre-wellorders on the reals,) carries measures (and extenders) of high Mitchell-order. Time permitting we will also indicate how to get a model of LSA from intermdiate strengthenings of the hypothesis.

  • Omer Ben-Neria + Martin Zeman (2 + 2 slots): On consistency strength of mutual stationarity with fixed uncountable cofinalities. Abstract: This is a joint work which is still in progress. We investigate the consistency strength of mutual stationarity at small singular cardinals of countable cofinality. The precise formulation for instance at ℵω reads that, given some uncountable γ=ℵk and a sequence Sn such that each Sn is a stationary subset of ℵn concentrating on ordinals of cofinality γ there is a stationary set of substructures X of Hθ (θ large given in advance) such that X ∩ ℵn ∈ Sn on a tail-end of n's. Lower bounds of this variant of mutual stationarity was investigated by several authors, predominantly Koepke and Welch. It turns out to have high consistency strength, and we are working on the core model induction in L(R) to obtain AD (we believe the actual consistency strength is much higher). In this talk I will present an argument which takes care of successor step of the core model induction. More precisely, an argument which shows the existence of an inner model with a Woodin cardinal. Here we use a covering type of argument, which is an approach different from the one used by Koepke and Welch, and our lower bound is an improvement of their result.

  • William Chan (2 slots): Definable Combinatorics at the First Uncountable Cardinal. Abstract: We work under the axiom of determinacy. By the strong partition property of ω1, for each ε ≤ ω1, there is a natural ε-partition measure on [ω1]ε. We will show the almost everywhere short function club uniformization which states that every relation R ⊂ [ω1]ω1 × club has a uniformization almost everywhere according to ω1-partition measure. This will be applied to show that every function from Φ : [ω1]ω1 → ω1 is continuous almost everywhere according to the ω1-partition measure. This also shows every partition of [ω1]ω1 into ω1 many pieces must have at least one piece of maximum cardinality [ω1]ω1. The club uniformization will also be used to study the stable theory of the ε-partition measures: In particular, it will be shown that for every ε ≤ ω1, L[f] is a model of GCH for almost all f according to the ε-partition measure. This is joint work with Jackson and Trang.

  • Gabriel Fernandes (1 slot): Inclusion modulo nonstationary. (This is joint work with A. Rinot and M. Moreno.) A classical theorem of Hechler asserts that the partial order <* on the reals is universal in the sense that for any sigma-directed poset P with no maximal element, there is a ccc forcing extension in which the reals ordered by <* contains a cofinal order-isomorphic copy of P. We prove a consistency result concerning the universality of the higher analogue (κκ,<S). We first verify that a refinement of ⋄, a diamond principle defined by Devlin, implies the universality of <S, and verify that such diamond is a consequence of local club condensation (lcc), lcc was defined by Friedman and Holy, it is an abstraction of a condensation property that many canonical inner models have. By Holy-Welch-Wu lcc can be forced by a set sized forcing.

  • Moti Gitik (1 slot): Around accumulation points and maximal sequences of indiscernibles. Abstract: Answering a question of Mitchell we show that a limit of accumulation points can be singular in K. It is shown also, that it is possible that there is no maximal sequence of indiscernibles for a former measurable, starting from o(κ)=κ.

  • Gabe Goldberg (1 slot): Generalizations of the Ultrapower Axiom. Abstract: The Ultrapower Axiom (UA) is a combinatorial principle motivated by inner model theory which provides an elegant setting for developing the part of large cardinal theory that concerns countably complete ultrafilters. On the other hand, many basic questions in the theory of extenders and strong cardinals remain independent of UA; e.g., is the least tall cardinal equal to the least strong cardinal? It is tempting to try to generalize UA to extenders in order to decide such questions, but in this talk, we will show that the simplest of these generalizations, called the Extender Power Axiom, is actually refuted by large cardinals. Our counterexample is closely related to the counterexamples to the Unique Branch Hypothesis. This is joint work with Hugh Woodin.

  • Steve Jackson (1 slot): Some results on the closure of Suslin cardinals under ultrapowers. Abstract: Assuming AD we show all regular Suslin cardinals are closed under ultrapowers. We also prove some addition results for the singular case.

  • Mitch Rudominer (2 slots): The mouse set theorem just past projective. Abstract: We will give a proof of the mouse set theorem for the pointclass Π1ω +1. This pointclass lies at the start of the next projective-like hierarchy beyond the projective. Π1ω +1 is a scaled pointclass and may be considered to be analoguous to Π13. Also Π1ω +1 = Π2(J2(R)).
    Let A be the set of reals which are Δ1ω +1 in a countable ordinal. A is analoguous to Q3. We will describe a mouse M and show that the set of reals of M is equal to A. This is analoguous to R ∩ M1 = Q3.
    M is the least mouse such that for all n ∈ ω there is a δn such that δn is a cardinal of M and is Woodin in M with respect to Σ1n subsets of δn. This mouse was isolated in 1993 and at that time we proved that R ∩ M ⊂ A, but we were not able to prove that A ⊂ M. Recently Woodin proved that A ⊂ M yielding the mouse set theorem.
    This talk will focus on Woodin's proof that if a real x is Δ1ω +1 in a countable ordinal, then x is in M.

  • Ernest Schimmerling (2 slots): Covering at limit cardinals of K. Abstract: Assume that there is no transitive class model with a Woodin cardinal. Let ν be a singular ordinal such that ν > ℵ2 and cf(ν) < |ν|. Suppose ν is a regular cardinal in K. Then ν is a measurable cardinal in K. Moreover, if cf(ν) > ω, then oK(ν) ≥ cf(ν). This is joint work with Bill Mitchell. Reference: Covering at limit cardinals of K by B. Mitchell and E. Schimmerling.

  • Ralf Schindler (4 slots): Tutorial on Covering. Abstract: The Covering Lemma for K, proven by Mitchell, Schimmerling, and Steel, says that below a Woodin cardinal, if κ is a cardinal above ℵ2, then the V-cofinality of κ+K is at least κ. Many results in inner model theory are actually in need of the argument proving the Covering Lemma rather than just its statement. Examples are: Above ℵ2, K is just the stack of "collapsing mice," and the K of any ω-closed model of ZFC is an iterate of K (both proven by the speaker). There are many contexts where it would be desirable to have a stronger sufficient condition for when a given premouse doesn't move in the comparison with K. (This also applies to Martin Zeman's talk at this meeting, so that this tutorial might be a helpful tool for his talk.) We intend to present the key steps in the Covering Lemma argument. References: The Covering Lemma up to a Woodin cardinal by Mitchell, Schimmerling and Steel, Iterates of the core model by the speaker, and Pcf theory and Woodin cardinals by Gitik, Shelah, and the speaker.

  • Farmer Schlutzenberg (3 slots): Varsovian models with infinitely many Woodin cardinals. Abstract: Suppose there is an OR-iterable least active mouse M which satisfies "there is a cardinal lambda which is a limit of strong cardinals and Woodin cardinals." Let M be given by iterating its top measure out of the universe. We will discuss the Varsovian model VM of M. We get that VM is an OR-iterable proper class strategy mouse with infinitely many Woodins, and the universe of VM equals the mantle of M and equals HODM[G], for sufficiently large collapse generics G over M. This builds on joint work with Sargsyan and Schindler.

  • Xianghui Shi (1 slot): On a question of Steel. Abstract: Under AD, there is no ω1 sequence of reals, and further there is no ω1 sequence of countable sets of reals. We show that the I0 counterparts of both statements are true under I0, namely, under I0(λ), in L(Vλ+1) there is no λ+ sequence of subsets of λ, as well as no λ+ sequence of subsets of the power set of λ which are each of size λ. The latter answers a question of Steel raised in the Arctic workshop of this year.

  • Benjamin Siskind (1 slot): Iterability for infinite stacks. Abstract: Farmer Schlutzenberg and John Steel showed that nice strategies for single normal trees have canonical extensions to strategies for finite stacks of normal trees. Schlutzenberg showed we can further extend to canonical strategies for infinite stacks. We will present this result using the framework of meta-iteration trees.

  • John Steel (3 slots): Mouse pairs and Suslin cardinals. A mouse pair is a pair (P,Σ) such that P is a premouse and Σ is an iteration strategy for P having a certain condensation property. The basic theorems of inner model theory (e.g. the Comparison Lemma and the Dodd-Jensen Lemma) are best stated as theorems about mouse pairs. One type of mouse pair can be used to analyze the HODs of models of ADR, leading to
    Theorem 0.1 Assume ADR + HPC; then HOD is a model of GCH.
    Here HPC stands for "hod pair capturing," a natural assumption concerning the existence of mouse pairs. An analysis of optimal Suslin representations for mouse pairs leads to
    Theorem 0.2 Assume ADR + HPC; then the following are equivalent:
    (a) δ is Woodin in HOD, and a cutpoint of the extender sequence of HOD,
    (b) δ = θ0 or δ = θα +1 for some α.
    Here the θα's are the Solovay sequence. Grigor Sargsyan introduced a refinement of the Solovay sequence in which ordinal definability from sets of reals is replaced by ordinal definablity from countable sequences of ordinals. Calling these ordinals ηα we have
    Theorem 0.3 Assume ADR HPC; then the following are equivalent:
    (a) δ is a successor Woodin in HOD,
    (b) δ = η0 or δ = ηα +1 for some α.
    This theorem was conjectured by Sargsyan.
    Reference: Mouse pairs and Suslin cardinals by J. Steel.

  • Nam Trang (4 slots): The strength of Sealing. Abstract: Sealing is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. LSA-over-UB is the statement that in all (set) generic extensions V[g] of V, there is some A such that L(A,R) satisfies LSA and L(A,R) contains every universally Baire set. Under a mild large cardinal hypothesis, we show that Sealing is equiconsistent with LSA-over-UB. As a consequence, Sealing is weaker than the theory "ZFC+there is a Woodin cardinal which is a limit of Woodin cardinals". Sealing's consistency being weak represents an obstruction to the current program of descriptive inner model theory. Going beyond this bound in core model induction applications seems challenging and requires us to construct third order objects (subsets of the universally Baire sets). This is joint work with G. Sargsyan.

  • Trevor Wilson I (1 slot): Rigidity and non-rigidity for labeled trees and for structures with one unary function and one unary relation. Abstract: A family of structures of the same similarity type is called rigid if there is no homomorphism from one structure in the family to another. For a similarity type with a binary relation symbol or two unary function symbols, the existence of arbitrarily large set-sized rigid families is a theorem of ZFC, and the non-existence of a proper class rigid family is equivalent to Vopenka's principle. Letting τ be the similarity type with one unary function symbol and one unary relation symbol, we show that non-rigidity for τ-structures is much weaker than this, and is equivalent to non-rigidity for 2-labeled trees. (Here an X-labeled tree is a tree of height ≤ ω with an arbitrary "labeling" function from its vertex set to the set X, and homomorphisms are required to preserve the root, the predecessor relation, the levels, and the labels.) Shelah proved that non-rigidity of cardinality-λ families of 2-labeled trees is equivalent to λ → (ω). We give a new, simple proof of Shelah's theorem using games, and apply it to show that non-rigidity of cardinality-λ families of τ-structures is also equivalent to λ → (ω). Moreover, we prove that non-rigidity of proper classes of 2-labeled trees and non-rigidity of proper classes of τ-structures are both equivalent to the generic Vopenka principle for countable languages, with natural strengthenings being equivalent to the generic Vopenka principle itself.

  • Trevor Wilson II (1 slot): The large cardinal strength of the weak and semi-weak Vopěnka principles We show that the Weak Vopěnka Principle (WVP) and Semi-Weak Vopěnka Principle (SWVP) are both equivalent to the large cardinal principle "Ord is Woodin," which says that for every class C there is a C-strong cardinal.

  • People who participated:

    Dominik Adolf (Bar Ilan Univ., Ramat Gan)
    Omer Ben-Neria (Jerusalem)
    David Casey (UC Berkeley)
    Justin Cavitt (Harvard Univ., Cambridge, MA)
    William Chan (Denton, TX)
    Gabriel Fernandes (Bar Ilan Univ., Ramat Gan)
    Elliot Glazer (Harvard Univ., Cambridge, MA)
    Gabriel Goldberg (Harvard Univ., Cambridge, MA)
    Takehiko Gappo (Rutgers Univ.)
    Moti Gitik (Tel Aviv)
    Steve Jackson (Denton, TX)
    Andreas Lietz (Münster)
    Mitch Rudominer (San Francisco)
    Ernest Schimmerling (CMU)
    Ralf Schindler (Münster)
    Farmer Schlutzenberg (Münster)
    Xianghui Shi (Beijing)
    Benjamin Siskind (UC Berkeley)
    John Steel (UC Berkeley, CA)
    Nam Trang (UC Irvine)
    Trevor Wilson (Miami Univ., Oxford OH)
    Martin Zeman (UC Irvine, CA)

    Housing information:

    Attached is a list of hotels that have UC rates. People who want to book at one of these hotels should mention that they are with our conference, and use Judie Filomeo's name to get the UC rate; her email address is judiew@math.berkeley.edu .

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