# Set theory seminar

# A countable ordinal definable set of reals without ordinal definable elements

In 2010, a question on MathOverflow asked whether it is possible that a countable ordinal definable set of reals has elements that are not ordinal definable. It is easy to see that every element of a finite ordinal definable set of reals is itself ordinal definable. Also it is consistent that there is an uncountable ordinal definable set of reals without ordinal definable elements. It turned out that the question for countable sets of reals was not known. It was finally solved by Kanovei and Lyubetsky in 2014, who showed, using a forcing extension by a finite-support product of Jensen reals, that it is consistent to have a countable ordinal definable set of reals without ordinal definable elements. In the talk, I will give full details of their argument.

An extended abstract is available on my blog here.

# Computable models of finite set theory

In 2001, Mancini and Zambella investigated computable models of fragments of set theory. They emplyed the Bernays-Rieger method of permutations to construct a computable model of finite set theory (i.e. $ZF_{fin}$ – the theory obtained from ZF by replacing the axiom of infinity by its negation). In 2009, Enayat, Schmerl and Visser showed how to build computable nonstandard models of this theory without the use of permutations. Furthermore, they demonstrated that in every computable model of ZF_{fin} every set (as viewed externally) has only finitely many elements (such models are called $\omega$-models). The corollaries of these results are, among others, that there are continuum-many nonisomorphic pointwise definable $\omega$-models of $ZF_{fin}$ and that $PA$ and $ZF_{fin}$ are not bi-interpretable. The purpose of the talk is to present proofs of the results of Macinini, Zambella and Enayat, Schmerl, Visser together with the corollaries.

# Coherent Systems of Finite Support Iterations

The method of matrix iterations was introduced by Blass and Shelah in their study of the dominating and the ultrafilter numbers. Since its appearance, the method has undergone significant development and applied to the study of many other cardinal characteristics of the continuum, including those associated to measure and category.

Recently, we were able to extend the technique of matrix iterations to a “third dimension” and so, evaluate the almost disjointness number in models where previously its value was not known. In addition, we obtain new constellations of the Cichon diagram (with up to seven distinct values). This is a joint work with Friedman, Mejia and Montoya.

# Regular Jónsson cardinals

For an infinite cardinal $\lambda$, we say that $\lambda$ is Jónsson if it satisfies the square bracket partition property $\lambda\to[\lambda]^{<\omega}_\lambda$. This means that, for every coloring $F\colon [\lambda]^{<\omega}\to\lambda$, there is some set $H\in [\lambda]^\lambda$ with the property that $\mathrm{ran}(F\upharpoonright [H]^{<\omega})\subsetneq\lambda$. It is rather well known that the consistency strength of "there is a Jónsson cardinal" lies above "$0^\sharp$ exists", but below the existence of a measurable. However, the question of what sorts of cardinals can be Jónsson has turned out to be rather difficult. The goal of this talk is to sketch a simplified proof of the result (due to Shelah) that if $\lambda$ is the least regular Jónsson cardinal, then $\lambda$ must be $\lambda\times\omega$-Mahlo. The advantage of the proof presented here is that the machinery employed can be easily generalized, and time permitting I would like to discuss how one might attempt to prove that such a cardinals must be greatly Mahlo.

# No seminars on Nov. 25

Friday, November 25 is the day after Thanksgiving, so there will be no seminars at the Graduate Center that day.

# Characterizing forcing extensions

I shall present a proof of a theorem of Bukovský from 1973 that characterizes the set-forcing extensions among all pairs of ZFC models $M\subseteq N$: these are precisely the pairs satisfying a uniform covering property. His result has recently resurfaced in the study of set-theoretic geology and can, for example, also be used to give a conceptual proof of (a version of) the intermediate model theorem.

# $V$ need not be a forcing extension of $\mathrm{HOD}$ or of the mantle

In 1972 Vopenka showed that $V$ is a union of set-generic extensions of $\mathrm{HOD}$ by establishing that every set in $V\setminus\mathrm{HOD}$ is set generic over $\mathrm{HOD}$. It is natural to consider whether that union can be replaced by a single forcing, possibly a proper class, over $\mathrm{HOD}$. In 2012 Friedman showed that $V$ is a class forcing extension of $\mathrm{HOD}$ by a partial order definable in $V$ – however, this leaves open the question of whether such a partial order can be defined in $\mathrm{HOD}$ itself. In this talk I will show that the qualifier ‘in $V$’ is necessary in Friedman’s theorem, by producing a model which is not class generic over $\mathrm{HOD}$ for any forcing definable in $\mathrm{HOD}$.

In the area of set theory known as set-theoretic geology, the mantle $M$ (the intersection of all grounds) is an inner model that enjoys a relationship to $V$ similar to that of $\mathrm{HOD}$, but ‘in the opposite direction’ – every set not in $M$ is omitted by a ground of $V$. Does it follow that we can build $V$ up over $M$ by iteratively adding those sets back in via forcing? In particular, does it follow that $V$ is a class forcing extension of $M$? The example produced in this talk will show that the answer is no – there is a model of set theory $V$ which is not a class forcing extension of $M$ by any forcing definable in $M$.

# Separating Class Determinacy

In his dissertation, Steel showed that both open determinacy and clopen determinacy have a reverse math strength of $\mathsf{ATR}_0$. In particular, this implies that clopen determinacy is equivalent to open determinacy (over a weak base theory). Gitman and Hamkins in recent work explored determinacy for class-sized games in the context of second-order set theory. They asked whether the analogue of Steel’s result holds: over $\mathsf{GBC}$, is open determinacy for class games equivalent to clopen determinacy for class games?

The answer, perhaps surprisingly, is no. In this talk I will present some recent results by Hachtman which answer the question of Gitman and Hamkins. We will see how to construct a transitive model of $\mathsf{GBC}$ which satisfies clopen class determinacy but does not satisfy open class determinacy.

# The isomorphism problem for rank-1 transformations

I’ll begin by describing two fascinating questions in ergodic theory, one being the isomorphism problem for measure-preserving transformations. I’ll survey some of the progress that has been made on these problems, including some partial solutions to the isomorphism problem for certain classes of measure-preserving transformations and some anti-classification results, stating that “nice” solutions to the isomorphism problem are impossible on other classes of measure-preserving transformations. Then I’ll discuss recent work I’ve done with Su Gao on the isomorphism problem for the class of rank-1 transformations, a generic class of measure-preserving transformations where the isomorphism relation is known to be, in some sense, well-behaved. (Background information ergodic theory will be introduced as needed.)

# No set theory seminar on October 21

Since many of the seminar members will be away, there will be no set theory seminar on October 21.

# No logic seminars on October 14

Since CUNY will follow a Tuesday schedule on Friday, October 14, we will not have any of the usual Friday logic seminars that day. However, there will be a talk by George Metcalfe at 4 pm, described below.

# The uniqueness of $\mathbb R$-supercompactness measures in ZFC

${\rm AD}_{\mathbb R}$ is a strengthening of the determinacy axiom that states that all games on the real numbers are determined. It is a Theorem of Solovay that under ${\rm ZF}+{\rm AD}_{\mathbb R}$ there is a fine, countably complete and normal filter on $P_{\omega_1}(\mathbb R)$, so $\omega_1$ is $\mathbb R$-supercompact. The exact consistency strength of the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact” is, however, weaker than the one of ${\rm ZF}+{\rm AD}_{\mathbb R}$.

One central interest of Inner Model Theory is to construct/find canonical models for theories extending ${\rm ZF}$. A natural question is, then, whether there is a canonical model for the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact”.

In this talk, we will discuss the consistency strength and minimal models of this theory. We will discuss the proof of the uniqueness of minimal models of this theory, under various appropriate hypotheses. And time permitting we will discuss the proof of the result that under ${\rm ZFC}$ there is at most one minimal model of this theory. This is joint work with Nam Trang.

# On the non-existence and definability of mad families

By an old result of Mathias, there are no mad families in the Solovay model constructed by the Levy collapse of a Mahlo cardinal. By a recent result of Törnquist, the same is true in the classical model of Solovay as well. In a recent paper, we show that ZF+DC+”there are no mad families” is actually equiconsistent with ZFC. I’ll present the ideas behind the proof in the first part of the talk.

In the second part of the talk, I’ll discuss the definability of maximal eventually different families and maximal cofinitary groups. In sharp contrast with mad families, it turns out that Borel MED families and MCGs can be constructed in ZF. Finally, I’ll present a general problem in Borel combinatorics whose solution should explain the above difference between mad and maximal eventually different families, and I’ll show how large cardinals must be involved in such a solution.

This is joint work with Saharon Shelah.

# The automorphic Borel amalgamation property

We show that, in many cases, there is a Borel reduction from the isomorphism relation on a given Fraïssé class to the conjugacy relation on the automorphism group of the Fraïssé limit. Hence, if the former is Borel complete, then so is the latter. The key property is a functorial, Borel form of amalgamation. All relevant notions about Borel redicibility and Fraïssé classes will be defined.

# Set-theoretic geology and the downward-directed grounds hypothesis: part II

I will continue presenting Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. See the main abstract at Set-theoretic geology and the downward directed ground hypothesis.

See my blog post about this talk.

# Set-theoretic geology and the downward-directed grounds hypothesis

Forcing is often viewed as a method of constructing larger models extending a given model of set theory. The topic of set-theoretic geology inverts this perspective by investigating how the current set-theoretic universe $V$ might itself have arisen as a forcing extension of an inner model. Thus, an inner model $W\subset V$ is a ground of $V$ if we can realize $V=W[G]$ as a forcing extension of $W$ by some $W$-generic filter $G\subset\mathbb Q\in W$. Reitz had inquired in his dissertation whether any two grounds of $V$ must have a common deeper ground. Fuchs, myself and Reitz introduced the downward-directed grounds hypothesis, which asserts a positive answer, even for any set-indexed collection of grounds, and we showed that this axiom has many interesting consequences for set-theoretic geology.

I shall give a complete detailed account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. This breakthrough result answers what had been for ten years the central open question in the area of set-theoretic geology and leads immediately to numerous consequences that settle many other open questions in the area, as well as to a sharpening of some of the central concepts of set-theoretic geology, such as the fact that the mantle coincides with the generic mantle and is a model of ZFC. I shall also present Usuba’s related result that if there is a hyper-huge cardinal, then there is a bedrock model, a smallest ground. I find this to be a surprising and incredible result, as it shows that large cardinal existence axioms have consequences on the structure of grounds for the universe.

See my blog post about this talk.

# Reflection of stationary sets, tree property and square

One of the most fruitful research area in set theory is the study of the so-called Reflections Principles. Roughly speaking, a reflection principle is a combinatorial statement of the following form: given a structure S (e.g. stationary sets, tree, graphs, groups …) and a property P of the structure, the principle establishes that there exists a smaller substructure of S that satisfies the same property P. There is a tension between large cardinals axioms and the axiom of constructibility V=L at the level of reflection: on the one hand, large cardinals typically imply reflection properties, on the other hand L satisfies the square principles which are anti-reflection properties. Two particular cases of reflection received special attention, the reflection of stationary sets and the tree property. We will discuss the interactions between these principles and a version of the square due to Todorcevic. This is a joint work with Menachem Magidor and Yair Hayut.

# A position in infinite chess with game value $\omega^4$

This talk is a more detailed version of the presentation that I gave on set theory day.

I present a position in infinite chess with game value $\omega^4$. Informally speaking, this means that one side can force a win in $\omega^4$ many moves, or that the game is equivalent to the game of counting down from $\omega^4$. This result, joint with Hamkins and Evans, improved on the previous best result of $\omega^3$.

# Normal Measures and Tall Cardinals

I will discuss the number of normal measures a non-$(\kappa + 2)$-strong tall cardinal $\kappa$ can carry, paying particular attention to the cases where $\kappa$ is either the least measurable cardinal or the least measurable limit of strong cardinals. This is joint work with James Cummings.

# Core model induction without the axiom of choice

We’ll discuss problems arising when trying to apply CMI in models where even the weakest forms of choice might fail. We’ll show how to deal with these problems in the particular case of a model in which all uncountable cardinals are singular.

# Generalized Laver tables II

We shall investigate several classes of left-distributive algebras that behave like algebras of elementary embeddings including permutative LD-systems, locally Laver-like LD-systems, and generalized Laver tables. In these algebras, there is a notion of a critical point, a composition operation, and the notion of equivalence up to a certain critical point. Furthermore, the locally Laver-like LD-systems are used to generate and classify generalized Laver tables. After discussing the general theory of these algebras, we shall show that there exists generalized Laver tables which cannot arise from algebras of elementary embeddings. We shall then give a framework that allows us to construct from rank-into-rank embeddings finite algebras that satisfy the distributivity identity $x*f(x_{1},…,x_{n})=f(x*x_{1},…,x*x_{n})$ where $(X,*)$ is a left-distributive algebra.

# wPFA

We isolate a new forcing axiom, ${\rm wPFA}$, which is strictly between ${\rm BPFA}$ and ${\rm PFA}$. ${\rm wPFA}$ is equiconsistent with a remarkable cardinal, it implies the failure of $\square_{\omega_1}$, but it is compatible with $\square_\kappa$ for all $\kappa \geq \omega_2$. This is part of joint work with J. Bagaria and V. Gitman.

# The Quandary of Quandles: The Borel Completeness of a Knot Invariant

We show that the isomorphism problems for left distributive algebras, racks, quandles, and keis are as complex as possible in the sense of Borel reducibility. These various kinds of algebraic structure are important for their connections with the theory of knots, links and braids, and in particular, Joyce showed that quandles could be used as complete invariants for tame knots. However, quandles have heuristically seemed to be unsatisfactory invariants. Our result confirms this view, showing that from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a harder problem.

# Singular in V, regular and non-measurable in HOD

Getting a model where $\kappa$ is singular in $V$ but measurable in ${\rm HOD}$ is somewhat straightforward however ensuring that $\kappa$ is regular but not measurable in ${\rm HOD}$ is a surprisingly more difficult problem. Magidor navigated around the issues and I will present his result starting with one measurable. His technique can be extended for set many cardinals.

# Why is Prikry forcing subcomplete?

Subcomplete forcing was introduced by Jensen as a class of forcings which do not add reals, but may change cofinalities to $\omega$, unlike proper forcing. In this talk I will show that Prikry forcing is subcomplete.

# Varsovian models

We compute the mantle of the minimal mouse with a Woodin and strong cardinal. It turns out to be the core model of the mouse. This is a joint work with Ralf Schindler.

# Beyond the Continuum: The Search for Higher Analogues of Martin’s Axiom

Martin’s axiom has been very successful in deciding numerous questions such as the Souslin hypothesis. Can we find higher analogues that, for example, decide the $\kappa$-Souslin hypothesis and allow us to meet $< 2^\kappa$ dense open sets? If one is careful enough and imposes closure conditions and stronger forms of the $\kappa$-cc, one obtains the principle BA$_\kappa$, independently discovered by Baumgartner, Laver and Shelah. BA$_\kappa$ shares many similarities with Martin’s axiom but decides the $\kappa$-Souslin hypothesis in the wrong way.

# Jónsson cardinals and club guessing

We say that a cardinal $\lambda$ is a Jónsson cardinal if it satisfies the following weak Ramsey-type property: given any coloring $F:[\lambda]^{<\omega}\to \lambda$ of the finite subsets of $\lambda$ in $\lambda$-many colors, there exists a set $H\in[\lambda]^\lambda$ such that the range of $F\upharpoonright [H]^{<\omega}$ is a proper subset of $\lambda$. One of the big driving forces present in early chapters Cardinal Arithmetic is an attempt to understand the combinatorial structure at and around Jónsson cardinals using scales and club guessing. The goal of this talk is to highlight the connection between Jónsson cardinals and the existence of certain sorts of club guessing ideals. Our focus will be on how club guessing ideals interact with Jónssonness at successors of singulars.

# No talks November 27

There will be no talks on November 27, the day after Thanksgiving.

# Subcomplete forcings

Subcomplete forcings are a class of forcings introduced by Jensen. These forcings do not add reals but may change cofinalities to $\omega$, unlike proper forcings. Examples of subcomplete forcings include Namba forcing, Prikry forcing, and any countably closed forcing. In this talk I will discuss some results concerning subcomplete forcing and the preservation of various properties of trees.

# Generalized Laver tables

The Laver tables are finite self-distributive algebras generated by one element that approximate the free left-distributive algebra on one generator if a rank-into-rank cardinal exists. We shall generalize the notion of a Laver table to a class of locally finite self-distributive algebraic structures with an arbitrary number of generators. These generalized Laver tables emulate algebras of rank-into-rank embeddings with an arbitrary number of generators modulo some rank. Furthermore, if there exists a rank-into-rank cardinal, then the free left-distributive algebras on any number of generators can be embedded in a canonical way into inverse limits of generalized Laver tables. As with the classical Laver tables, the reduced generalized Laver tables can be given an associative operation that is analogous to the composition of elementary embeddings and satisfies the same identities that algebras of elementary embeddings are known to satisfy. Furthermore, the notion of the critical point also holds in these generalized Laver tables as well even though generalized Laver tables are locally finite or finite. While the only classical Laver tables are the tables of cardinality $2^{n}$, the finite generalized laver tables occur much more frequently and many generalized Laver tables can be constructed from the classical Laver tables. We shall give some results that allow one to quickly compute the self-distributive operation in a certain class of generalized Laver tables.

Here are the slides.

# The rearrangement number

The Riemann rearrangement theorem states that a convergent real series $sum_n a_n$ is absolutely convergent if and only if the value of the sum is invariant under all rearrangements $sum_n a_{p(n)}$ by any permutation $p$ on the natural numbers; furthermore, if a series is merely conditionally convergent, then one may find rearrangements for which the new sum $sum_n a_{p(n)}$ has any desired (extended) real value or which becomes non-convergent. In recent joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson, based on an exchange in reply to a Hardy’s MathOverflow question on the topic, we investigate the minimal size of a family of permutations that can be used in this manner to test an arbitrary convergent series for absolute convergence. Specifically, we define the rearrangement number $rr$, a new cardinal characteristic of the continuum, to be the smallest cardinality of a set $P$ of permutations of the natural numbers, such that if a convergent real series $sum_n a_n$ remains convergent to the same value after any rearrangement $sum_n a_{p(n)}$ by a permutation $p$ in $P$, then it is absolutely convergent. The corresponding rearrangement number for sums, denoted rr_Sigma, is the smallest cardinality of a family $P$ of permutations, such that if a series $sum_n a_n$ is conditionally convergent, then there is some rearrangement $sum_n a_{p(n)}$, by a permutation $p$ in $P$, for which the series converges to a different value. We investigate the basic properties of these numbers, and explore their relations with other cardinal characteristics of the continuum. Our main results are that b≤ rr≤ non(M), that d≤ rr_Sigma, and that b≤ rr is relatively consistent.

# Nonstandard and relative set theories.

These theories axiomatize a universe of sets that can have nonstandard elements such as infinitesimals. The nonstandard set theory BST [respectively, the relative set theory GRIST] extends the language of ZFC by a unary predicate “x is standard” [respectively, by a binary predicate “x is standard relative to y”].

**Theorem.** Every model M of ZFC has an extension to a model of BST [respectively, GRIST] in which M is the universe of standard sets. If M is countable, then the extension is unique, modulo an isomorphism that fixes standard sets.

**Corollary**. BST [respectively, GRIST] is conservative and complete over ZFC.

I will describe some ideas used to prove these results, in particular, the technique of internally iterated ultrapowers.

# Tree representations from very large cardinals

We will discuss the propagation of certain tree representations in the presence of very large cardinals. These tree representations give generic absoluteness results and have structural consequences in the area of generalized descriptive set theory. In fact these representations give us a method for producing models of strong determinacy axioms.

# Layered partial orders

If $\kappa$ is a regular uncountable cardinal and $\mathbb{P}$ is a partial order, we say that $\mathbb{P}$ is $\kappa$ stationarily layered iff the set of regular suborders of $\mathbb{P}$ is stationary in $[\mathbb{P}]^{<\kappa}$. This is a strong form of the $\kappa$-chain condition, and in fact implies that $\mathbb{P}$ is $\kappa$-Knaster. I will discuss two recent applications involving layered posets:

(1) a new characterization of weak compactness: a regular $\kappa$ is weakly compact iff every $\kappa$-cc poset is $\kappa$ stationarily layered. This is joint work with Philipp Luecke.

(2) a general theorem about preservation of $\kappa$-cc under universal Kunen-style iterations.

# The Mitchell order for Ramsey cardinals

The usual Mitchell relation on normal measures on a measurable cardinal $\kappa$ orders the measures based on the degree of measurability that $\kappa$ retains in their respective ultrapowers. We shall examine the analogous ordering of appropriate witnessing objects for Ramsey (and Ramsey-like) cardinals. It turns out that the resulting order is well-behaved and its degrees neatly stratify the large cardinal hierarchy between Ramsey, strongly Ramsey, and super Ramsey cardinals. We also give a soft killing argument for this notion of Mitchell rank.

This is joint work with Victoria Gitman and Erin Carmody.

# Is there a least transitive model of Kelley-Morse Set Theory?

It’s well-known that there is a least transitive model of ZFC. $L_\alpha$, where $\alpha$ is the least ordinal which is the $\mathrm{Ord}$ of a model of set theory is contained in every transitive model of ZFC. With a little bit of effort, one can extend this to see that there is a least transitive model of GBC; its first-order part is $L_\alpha$ and its second-order part is the definable classes. Can we extend this result further to get that there is a least transitive model of KM? The purpose of this talk is to answer this question in the negative.

# No talks on September 25

On Friday, September 25, CUNY will follow a Tuesday schedule. Therefore, no logic seminars will meet that day.

# Boolean ultrapowers and the Bukovsky-Dehornoy phenomenon

I will present a criterion for when an ultrafilter on a Boolean algebra gives rise to the Bukovsky-Dehornoy phenomenon, namely that the intersection of all intermediate ultrapowers is equal to the the Boolean model. Time permitting, I will show that the Boolean algebras of Prikry and Magidor forcing satisfy the strong Prikry property, and that these forcings come with a canonical imitation iteration whose limit model is the Boolean ultrapower by a very canonical ultrafilter on their respective Boolean algebras.

# Admissible Covers and Compactness Arguments for Ill-founded Models of Set Theory

The Barwise compactness theorem is a powerful tool, allowing one to prove many interesting results which cannot be gotten just from the ordinary compactness theorem. For example, it can be used to show that every countable transitive model of set theory has an end extension which is a model of $V = L$. However, the Barwise compactness theorem only applies to transitive sets. What are we to do if we want to have compactness arguments for ill-founded models of set theory?

This is where Barwise’s notion of the admissible cover of a (possibly ill-founded) model of set theory comes in. In this talk, we will see how to construct admissible covers and how they can be used to extend compactness arguments to the ill-founded case.

This talk is a prequel of sorts to my next talk in this seminar. Some of the results discussed in this talk will play a crucial roll in the arguments there.

# Ultrafilters and nonstandard methods in combinatorics of numbers

In certain areas of Ramsey theory and combinatorics of numbers, diverse non-elementary methods are successfully applied, including ergodic theory, Fourier analysis, (discrete) topological dynamics, algebra in the space of ultrafilters. In this talk I will survey some recent results that have been obtained by using tools from mathematical logic, namely ultrafilters and nonstandard models of the integers.

On the side of Ramsey theory, I will show how the hypernatural numbers of nonstandard analysis can play the role of ultrafilters, and provide a convenient setting for the study of partition regularity problems of diophantine equations. About additive number theory, I will show how the methods of nonstandard analysis can be used to prove density-dependent properties of sets of integers. A recent example is the following theorem: If a set $A$ of natural numbers has positive upper asymptotic density then there exists infinite sets $B$, $C$ such that their sumset $C+B$ is contained in the union of $A$ and a shift of $A$. (This gives a partial answer to an old question by Erdős.)

The slides are here.

# Split Principles

In this talk I will introduce various “Split Principles”, which, if they hold, posit the existence of a sequence which in some sense “splits” any large set into two unbounded pieces. We will see that the failure of a particular split principle to hold tends to characterize some large cardinal property; in particular weak compactness, ineffability, measurability and supercompactness can each be characterized in terms of the failure of a split principle. The initial idea of a split principle came about recently in the joint work of Fuchs, Gitman, and Hamkins. The content of the talk is the result of exploring the idea further with Gunter Fuchs.

# Constructing joint diamonds from a single diamond

A joint diamond sequence on a cardinal $\kappa$ is a collection of $\diamondsuit_\kappa$ sequences which coheres in the sense that any collection of subsets of $\kappa$ may be guessed on stationary sets in some normal uniform filter on $\kappa$. This is the direct translation of joint Laver diamonds to smaller $\kappa$ which have no suitable elementary embeddings. In this talk I will show that, as opposed to the large cardinal case, joint diamond sequences simply exists whenever $\diamondsuit_\kappa$ holds.

# Deciphering the HOD Dichotomy: Part II

I will continue to attempt to present my version of the notes of Woodin’s talk at the Appalachian Set Theory seminar on his paper, the HOD Dichotomy, including the HOD conjecture. Principally, I hope to present how weak extender models relate to the HOD conjecture. I will also present the initial part of new results that were inspired by the HOD conjecture, in particular, by failures of the cover property.

# Diamond* Coding

Coding information into the structure of the universe is a forcing technique with many applications in set theory. To carry out it out, we a need a property that: i) can be easily switched on or off at (e.g.) each regular cardinal in turn, and ii) is robust with regards both to small and to highly-closed forcing. GCH coding, controlling the success or failure of the GCH at each cardinal in turn, is the most widely used, and for good reason: there are simple forcings that turn it on and off, and it is easily seen to be unaffected by small or highly-closed forcing. However, it does have limitations – most obviously, GCH coding is of necessity incompatible with the GCH itself. In this talk I will present an alternative coding using the property Diamond*, a variant of the classic Diamond. I will discuss Diamond* and demonstrate that it satisfies the requirements for coding while preserving the GCH.

Although the basic techniques for controlling Diamond* have been known for some time, to my knowledge the first use of Diamond* as a coding axiom was by Andrew Brooke-Taylor in his work on definable well-orders of the universe. I will follow the excellent exposition presented in his dissertation.

# Dissertation Defense: Force to change large cardinals

This will be the dissertation defense of the speaker. There will be a one-hour presentation, followed by questions posed by the dissertation committee, and afterwards including some questions posed by the general audience. The dissertation committee consists of Joel David Hamkins (supervisor), Gunter Fuchs, Arthur Apter, Roman Kossak and Philipp Rothmaler.

# When is a Boolean ultrapower an ultrapower?

The Boolean ultrapower construction is a natural generalization of the classical ultrapower construction, but the Boolean ultrapower uses an ultrafilter on a complete Boolean algebra instead of a set. It was initially unknown as to whether in ZFC there exists a Boolean ultrapower which is not always isomorphic to a classical ultrapower. This problem was resolved in 1976 by Bernd and Sabine Koppelberg who constructed a Boolean ultrapower which is not an ultrapower. On the other hand, there does not seem to be any reference in the mathematical literature to atomless Boolean ultrapowers which are isomorphic to classical ultrapowers.

We shall first generalize the notion of a Boolean ultrapower to the notion of a BPA-ultrapower which is in a sense the most general ultrapower construction. Then by applying a result of Joel David Hamkins which characterizes the Boolean ultrapowers which are classical ultrapowers, we shall investigate examples of Boolean ultrapowers which are not classical ultrapowers as well as Boolean ultrapowers which are classical ultrapowers. For instance, I claim that under GCH every complete atomless Boolean algebra has an ultrafilter which gives rise to a Boolean ultrapower which is not a classical ultrapower. On the other hand, using the Keisler-Shelah isomorphism theorem, we may construct Boolean ultrapowers in ZFC on a fairly general class of Boolean algebras which are classical ultrapowers.

# The boldface resurrection axioms

Friday, April 10, 2015

# Spring break

CUNY’s spring vacation is April 3-11, 2015. Therefore, no seminars will meet at the Graduate Center on April 3, nor on April 10.

# Harrington’s Principle and remarkable cardinals

It is open whether $\Pi^1_1$ determinacy implies the existence of $0^{\#}$ in 3rd order arithmetic, call it $Z_3$. We compute the large cardinal strength of $Z_3$ plus “there is a real $x$ such that every $x$-admissible is an $L$-cardinal.” This is joint work with Yong Cheng.

# Generic I0 at $\aleph_\omega$

It is common practice to consider the generic version of large cardinals defined with an elementary embedding, but what happens when such cardinals are really large? The talk will concern a form of generic I0 and the consequences of this over-the-top hypothesis on the “largeness” of the powerset of $\aleph_\omega$. This research is a result of discussions with Hugh Woodin.

# Force to change large cardinal strength

Suppose $kappain V$ is a cardinal with large cardinal property $A$. In this talk, I will present several theorems which exhibit a notion of forcing $mathbb P$ such that if $Gsubseteq mathbb P$ is $V$-generic, then the cardinal $kappa$ no longer has property $A$ in the forcing extension $V[G]$, but has as many large cardinal properties below $A$ as possible. I will also introduce new large cardinal notions and degrees for large cardinal properties.

This talk is the speaker’s dissertation defense.

# Generalized Baire spaces and closed Maximality Principles

Given an uncountable regular cardinal $\kappa$, the generalized Baire space of $\kappa$ is set ${}^\kappa\kappa$ of all functions from $\kappa$ to $\kappa$ equipped with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than $\kappa$.

A subset of this space is $\mathbf{\Sigma}^1_1$ (i.e. a projection of a closed subset of ${}^\kappa\kappa\times{}^\kappa\kappa$) if and only it is definable over $\mathrm{H}(\kappa^+)$ by a $\Sigma_1$-formula with parameters. This shows that the class of $\mathbf{\Sigma}^1_1$-subsets contains a great variety of set-theoretically interesting objects. Moreover, it is known that many basic and interesting questions about sets in this class are not decided by the axioms of $\mathrm{ZFC}$ plus large cardinal axioms.

In my talk, I want to present examples of extensions of $\mathrm{ZFC}$ that settle many of these questions by providing a nice structure theory for the class of $\mathbf{\Sigma}^1_1$-subsets of ${}^\kappa\kappa$. These forcing axioms appear in the work of Fuchs, Leibman, Stavi and Väänänen. They are variations of the *maximality principle* introduced by Stavi and Väänänen and later rediscovered by Hamkins.

# Rather classless models of set theory and second-order set theory

A model $M$ of ZFC is rather classless if every class of $M$ all of whose bounded initial segments are in $M$ is definable in $M$. In this talk, we will construct rather classless end extensions for every countable model of set theory. As an application of this construction, we will see that there are models of ZFC with precisely one extension to a model of GBC and that there are models of set theory which admit no extension to a model of GBC. If time permits, we will look at some related constructions with models of KM + the axiom schema of class choice.

# Embeddings of the universe into the constructible universe, current state of knowledge

I shall describe the current state of knowledge concerning the question of whether there can be an embedding of the set-theoretic universe into the constructible universe. The main question is: can there be an embedding $j:Vto L$ of the set-theoretic universe $V$ into the constructible universe $L$, when $Vneq L$? The notion of *embedding* here is merely that $xin y$ if and only if $j(x)in j(y)$, and such a map need not be elementary nor even $Delta_0$-elementary. It is not difficult to see that there can generally be no $Delta_0$-elementary embedding $j:Vto L$, when $Vneq L$. Nevertheless, the question arises very naturally in the context of my previous work on the embeddability phenomenon, which shows that every countable model $M$ does admit an embedding $j:Mto L^M$ into its constructible universe. More generally, any two countable models of set theory are comparable; one of them embeds into the other. Indeed, one model $langle M,in^Mrangle$ embeds into another $langle N,in^Nrangle$ just in case the ordinals of the first $text{Ord}^M$ order-embed into the ordinals of the second $text{Ord}^N$. In these theorems, the embeddings $j:Mto L^M$ are defined completely externally to $M$, and so it was natural to wonder to what extent such an embedding might be accessible inside $M$. Currently, the question remains open, but we have some partial progress, settling it in a number of cases.

This is joint work of myself, W. Hugh Woodin, Menachem Magidor, with contributions also by David Aspero, Ralf Schindler and Yair Hayut. See more information at the links below:

Blog post for this talk | Related MathOverflow question | Article

# Remarkable Laver functions

Since Laver defined and used a Laver function to show that supercompact cardinals can be made indestructible by all $\lt\kappa$-directed closed forcing, Laver-like functions have been defined for various large cardinals and used for lifting embeddings in indestructibility arguments. Laver-like functions are also inherently interesting as guessing principles with affinity to $\diamondsuit$. Supposing that a large cardinal $\kappa$ can be characterized by the existence of some kind of embeddings, a Laver-like function $\ell:\kappa\to V_\kappa$ has, roughly speaking, the property that for any set $a$ in the universe, there is an embedding $j$ of the type characterizing the cardinal such that $j(\ell)(\kappa)=a$. Although Laver-like functions can be forced to exist for almost any large cardinal, only a few large cardinals including supercompact, strong, and extendible, have them outright. I will define the notion of a *remarkable Laver* function for a remarkable cardinal and show that every remarkable cardinal has a remarkable Laver function. Remarkable cardinals were introduced by Ralf Schindler who showed that a remarkable cardinal is precisely equiconsistent with the property that the theory of $L(\mathbb R)$ is absolute for proper forcing. Time permitting, I will show how the existence of remarkable Laver functions is used in demonstrating indestructibility for remarkable cardinals. This is joint work with Yong Cheng. An extended abstract can be found here.

# The tall and measurable cardinals can coincide on a proper class

Starting from an inaccessible limit of strong cardinals, we force to construct a model containing a proper class of measurable cardinals in which the tall and measurable cardinals coincide precisely. This is joint work with Moti Gitik which extends and generalizes an earlier result of Joel Hamkins.

# Critical sequences of rank-to-rank embeddings and a tower of finite left distributive algebras: Part II

The speaker will continue to discuss the properties of rank-into-rank embeddings and their connections to the study of the tower of finite left-distributive algebras known as Laver Tables.

# Producing $M_n^{\#}$ from Boldface Level-wise Projective Determinacy

Projective determinacy is the statement that for certain infinite games, where the winning condition is projective, there is always a winning strategy for one of the two players. It has many nice consequences which are not decided by ZFC alone, e.g. that every projective set of reals is Lebesgue measurable. An old so far unpublished result by W. Hugh Woodin is that one can derive specific countable iterable models with Woodin cardinals, $M_n^{\#}$, from this assumption. Work by Itay Neeman shows the converse direction, i.e. projective determinacy is in fact equivalent to the existence of such models. These results connect the areas of inner model theory and descriptive set theory. We will give an overview of the relevant topics in both fields and, if time allows, sketch a proof of the result that for the odd levels of the projective hierarchy boldface $\Pi^1_{2n+1}$-determinacy implies the existence of $M_{2n}^{\#}(x)$ for all reals $x$.

# A nice model for cardinal characteristics at a supercompact *κ*

In a throw-away comment in a relatively recent preprint, Garti and Shelah state that using the technique of Dzamonja and Shelah, one can start with a model of set theory containing a supercompact cardinal *κ*, and force to obtain a model in which *κ* remains supercompact, * 2^{κ}* is large, but the ultrafilter number at

*κ*is only

*κ*. I will present this construction, and with it further results from joint work with Vera Fischer and Diana Montoya pinning down many other generalized cardinal characteristics at

^{+}*κ*in the resulting model.

# Does the HOD dichotomy preclude a model of a proper class of supercompact cardinals which are not HOD-supercompact?

I will be presenting my version of the notes of Woodin’s talk at the Appalachian Set Theory seminar on his paper, the HOD Dichotomy. In particular I will be discussing Woodin’s version and definition of a model of set theory being “far from HOD”. I will also be discussing how this relates to some possible future results (see title) as well as a recent result of Cheng,Friedman and Hamkins, which seem on their face to contradict Woodin’s premise.

# Critical sequences of rank-to-rank embeddings and a tower of finite left distributive algebras

In the early 1990’s Richard Laver discovered a deep and striking correspondence between critical sequences of rank-to-rank embeddings and finite left distributive algebras on integers. Each $A_n$ in the tower of finite algebras (commonly called the Laver Tables) can be defined purely algebraically, with no reference to the elementary embeddings, and yet there are facts about the Laver tables that have only been proven from a large cardinal assumption. We present here some of Laver’s foundational work on the algebra of critical sequences of rank-to-rank embeddings and related algebraic work of Laver’s and the author’s, describe how the finite algebras arise from the large cardinal embeddings, and mention several related open problems.

# News on the Solid Core

Set-theoretic geology, a line of research jointly created by Hamkins, Reitz and myself, introduced some inner models which result from inverting forcing in some sense. For example, the mantle of a model of set theory V is the intersection of all inner models of which V is an extension by set-forcing. It was an initial, naive hope that one might arrive at a model that is in some sense canonical, but one of the main results on set-theoretic geology is that this is not so: every model of set theory V has a class forcing extension V[G] so that the mantle, as computed in V[G], is V. So quite literally, the mantle of a model of set theory can be anything.

In an attempt to arrive at a concept that fits in with the general spirit of set-theoretic geology, but that stands a chance of being canonical, I defined a set to be solid if it cannot be added to an inner model by set-forcing, and I termed the union of all solid sets the “solid core”.

I will present some results on the solid core which were obtained in recent joint work with Ralf Schindler, and which show that the solid core indeed is a canonical inner model, assuming large cardinals (more precisely, if there is an inner model with a Woodin cardinal), but that it is not as canonical as one might have hoped without that assumption.

# Scott’s problem for models of ZFC

Scott’s problem asks which Scott sets can be represented as standard systems of models of Peano arithmetic. Knight and Nadel proved in 1982 that every Scott set of cardinality $\le \omega_1$ is the standard system of some model of PA. Little progress has been made since then on the problem. I will present a modification of these results to models of ZFC and show that every Scott set of cardinality $\le \omega_1$ is the standard system of some model of ZFC. This talk will constitute the speaker’s oral exam.

# Mutual Stationarity and Prikry-type forcings

Mutual stationarity is a property first introduced by Foreman and Magidor to study saturation properties of nonstationary ideals. Given a sequence $\langle\kappa_i : i < \lambda\rangle$ of regular cardinals, a sequence $\langle S_i: i < \lambda\rangle$ with $S_i \subseteq \kappa_i$ stationary for every $i$, is mutually stationary iff there are stationarily many subsets $A \subseteq \sup_{i < \lambda} \kappa_i$ s.t. $\sup(A \cap \kappa_i) \in S_i$ for all $i$ with $\kappa_i \in A$. Consider this second property of a sequence $\langle\kappa_i : i < \lambda\rangle$: there is a forcing $P$ that changes $\text{cof}(\kappa_i)$ to $\eta_i$ without changing cofinalities or cardinalites of ordinals below $\inf{\kappa_i : i < \lambda}$. We want to discuss how, and why, these properties are related.

# Elementary epimorphisms between models of set theory

This talk concerns joint work with Robert S. Lubarsky.

Elementary epimorphisms were introduced by Philipp Rothmaler. A surjective homomorphism f: M –> N between two model-theoretic structures is an elementary epimorphism if and only if every formula with parameters satisfied by N is satisfied in M using a preimage of those parameters.

Philipp asked me whether nontrivial elementary epimorphisms between models of set theory exist. We answer this question in the negative for fully elementary epimorphisms between models of ZFC, but in the positive under weaker assumptions.

In particular, we show that every Pi_1-elementary epimorphism between models of ZF is an isomorphism. On the other hand, nonisomorphic Sigma_1-elementary epimorphisms between models of ZF can be constructed, as can fully elementary epimorphisms between models of ZFC^-.

# When does every definable set have a definable member?

Although the concept of `being definable’ is not generally expressible in the language of set theory, it turns out that the models of ${rm ZF}$ in which every definable nonempty set has a definable element are precisely the models of $V={rm HOD}$. Indeed, $V={rm HOD}$ is equivalent to the assertion merely that every $Pi_2$-definable set has an ordinal-definable element. Meanwhile, this is not true in the case of $Sigma_2$-definability, because every model of ZFC has a forcing extension satisfying $Vneq{rm HOD}$ in which every $Sigma_2$-definable set has an ordinal-definable element. This is joint work with François G. Dorais and Emil Jeřábek, growing out of some questions on MathOverflow, namely,

- Definable collections without definable members
- A question asked by Ashutosh five years ago, in which François and I gradually came upon the answer together.
- Is it consistent that every definable set has a definable member?
- A similar question asked last week by (anonymous) user38200
- Can $Vneq{rm HOD}$ if every $Sigma_2$-definable set has an ordinal-definable member?
- A question I had regarding the limits of an issue in my answer to the previous question.

In this talk, I shall present the answers to all these questions and place the results in the context of classical results on definability, including a review of basic concepts for graduate students.

# No seminars on Sept. 26 or October 3

CUNY will have holidays on two consecutive Fridays, September 26 and October 3, 2014, so the Logic Workshop and other seminars will not meet on those days.

# Joint Laver diamonds

A Laver diamond for a given large cardinal $\kappa$ is a function $\ell$, defined on $\kappa$, such that $j(\ell)(\kappa)$ can take any reasonable value, where $j$ is a relevant large cardinal embedding. A sequence of such functions is called jointly Laver or a joint Laver diamond if they can be made to take any given sequence of such values at the same time via a single embedding. In the talk we will consider questions about when such sequences outright exist, when their existence is equiconsistent with and when their existence is consistency-wise strictly stronger than the large cardinal in question.

# An alternate proof of the Halpern-Läuchli Theorem in one dimension

I will present a new proof of the strong subtree version of the Halpern-Läuchli Theorem, using an ultrafilter on $\omega$. The one dimensional Halpern-Läuchli Theorem states that for every finite partition of an infinite, finitely branching tree $T$, there is one piece $P$ of the partition and a strong subtree $S$ of $T$ such that $S \subseteq P$. This will cover the one dimensional case, with hopes that the proof can be extended to cover a product of trees.

# Local Ramsey theory: an abstract approach

This talk is about an abstract version of the notion of semi-selective co-ideal for subsets of a topological Ramsey space. This version is useful to characterize the corresponding generalization of the local Ramsey property in “topological terms”. We will also talk about forcing notions related to this abstract version of semi-selectivity, generalizing those related to Ellentuck’s space, and we will comment on some applications.

# The Laver Preparation

I will present the result of Laver demonstrating that the existence of a supercompact cardinal implies the existence of a Laver function, and using this to construct a model with a supercompact cardinal k which remains supercompact in any k-directed-closed forcing extension.

# Additional remarks on remarkable cardinals

This is a continuation of the earlier Introduction to remarkable cardinals lecture. The speaker will continue to discuss the various equivalent characterizations of remarkable cardinals and their relationship to other large cardinal notions.

# Namba-like Forcings at Successors of Singular Cardinals

Following up on Peter Koepke’s Logic Workshop lecture of March 22, 2013, I will discuss Namba-like forcings which either exist or can be forced to exist at successors of singular cardinals.

# Definability versus Algebraicity

In a recent paper, Hamkins and Leahy introduce the concept of algebraicity in the set theoretic context. Thus, a set is algebraic in a model of set theory if it belongs to a finite set definable in the model. Clearly, algebraicity is a weak form of definability, and it can be varied in similar ways as definability, for example by allowing parameters. While the authors showed that the class of hereditarily ordinal algebraic sets is equal to the class of hereditarily ordinal definable sets, many fundamental questions on the relationship between algebraicity and definability were open: in particular, the question whether these concepts can be different in a model of set theory. I will show how to produce models of set theory in which there are algebraic sets that are not ordinal definable, and construct a model in which there is a set which is internally algebraic (i.e., which belongs to a definable set the model believes to be finite), but not externally.

# Structural Connections Between a Forcing Class and its Modal Logic

This talk is on recent work with Joel Hamkins and Benedikt Loewe on ways in which finite-frame properties of specific modal logics can be combined with assertions in ZFC to show that these modal logics are related to those which arise from interpreting Gamma-forcing extensions of a model of ZFC as possible worlds of a Kripke model, where Gamma can be any of several classes of notions of forcing.

# On the independence of Borel’s Conjecture

In 1919 Borel introduced the notion of a strong measure zero set and stated what has become known as Borel’s conjecture (BC): every strong measure zero set of reals is countable. Sierpiński soon proved (1928) that CH implies the failure of BC. However, a proof for the consistency of BC with ZFC would have to wait for the development of more powerful tools. In 1976, Laver used an iterated forcing argument to produce a model of ZFC + BC. I will present an exposition of these classical results. Time permitting, I will sketch some analogous results for the dual Borel Conjecture, the category analogue of BC.

# Reflecting I_0

We will present an argument for reflecting the large cardinal axiom I_0 from marginally stronger large cardinals. This will involve presenting some of the theory of inverse limits, which R. Laver first studied in the context of reflecting large cardinals at this level. Along the way we will see many local reflection results below I_0 and state a strong form of reflection which is useful in other contexts.

# A proof of the relative consistency of PFA

I will use a supercompact cardinal to force the Proper Forcing Axiom (PFA). I will follow Baumgartner’s original argumet, but will use lottery sums instead of a Laver function.

# Introduction to remarkable cardinals

Ralf Schindler introduced remarkable cardinals because he discovered that they are precisely equiconsistent with the statement that the theory of $L(\mathbb R)$ is absolute for proper forcing. The statement that the theory of $L(\mathbb R)$ is absolute for all set forcing is closely related to whether $L(\mathbb R)\models {\rm AD}$. In contrast, remarkable cardinals sit relatively low in the large cardinal hierarchy; for instance, they are downward absolute to $L$. I will discuss the various equivalent characterizations of remarkable cardinals due to Schindler and show where the remarkable cardinals fit into the large cardinal hierarchy using results due to Schindler, Philip Welch and myself.

An extended abstract can be found here.

# The consistency strength of PFA for posets preserving aleph_2 or aleph_3

While the consistency strength of PFA is quite high in the large cardinal hierarchy, it is reasonable to expect that tame fragments of PFA should require much weaker assumptions. I will present an argument of Hamkins and Johnstone (2008) which shows the consistency of PFA for posets preserving aleph_2 or aleph_3 from a strongly unfoldable cardinal, a much smaller large cardinal which is, roughly speaking, to strongness (or supercompactness) as weak compactness is to measurability.

# Covering, core model induction, and hod mice (Part II)

We will discuss recent covering principles and show how they can be used to derive strength from failure of square.

# Covering, core model induction and hod mice

We will discuss recent covering principles and show how they can be used to derive strength from failure of square.

# Normal Measures and Strongly Compact Cardinals

I will discuss the question of the possible number of normal

measures a non-kappa + 2 strong strongly compact cardinal kappa

can carry. This is part of a joint project with James Cummings.

# Large cardinals need not be large in HOD

I will demonstrate that a large cardinal need not exhibit its large cardinal nature in HOD. I shall begin with the example of a measurable cardinal that is not measurable in HOD, and then afterward describe how to force more extreme examples, such as a model with a supercompact cardinal, which is not weakly compact in HOD. This is very recent joint work with Cheng Yong.

# Welcome to the Set Theory seminar Spring 2014

The set theory seminar will be coordinated in Spring 2014 by Thomas Johnstone.

Tom pointed out that it’s been about ten years that we have been having our weekly Set Theory seminar at CUNY, starting with a few of Joel Hamkins’s graduate students who met on a weekly basis (George, Jonas, Victoria, and himself), and we’ve had many interesting seminar meetings since then.

At this point the schedule is wide open, so please let Tom know if you’d be interested to give a talk.

# A non-standard explanation of Reverse Mathematics

Reverse Mathematics (RM) is a program in the Foundations

of Mathematics initiated around 1975 by Harvey Friedman. We

discuss the aims and results of RM, in particular the “Big Five”

phenomenon. We consider non-standard explanations and

nonstandard robustness results for the Big Five phenomenon.

This is joint work with Damir Dzhafarov.

# Bagaria’s characterization of bounded forcing axioms in terms of generic absoluteness

Goldstern and Shelah (1995) introduced the class of bounded forcing axioms, that is forcing axioms for families of antichains of bounded size. For example, the bounded proper forcing axiom ${\rm BPFA}$ asserts that for any proper forcing notion $\mathbb{P}$ and any collection $D$ of at most $\aleph_1$ many maximal antichains in $\mathbb{P}$, each of size at most $\aleph_1$, there is a filter on $\mathbb{P}$ meeting each antichain in $D$. The speaker will present a theorem of Joan Bagaria (2000) that characterizes bounded forcing axioms in terms of generic absoluteness: for instance, Bagaria’s result shows that ${\rm BPFA}$ is equivalent to the assertion that if a $\Sigma_1$ sentence of the language of set theory with parameters of hereditary size at most $\aleph_1$ is true in some proper forcing extension, then it is already true in the ground model.

# A Jónsson $\omega_1$-like model of set theory

A first-order structure of cardinality $\kappa$ is said to be Jónsson if it has no proper elementary substructure of cardinality $\kappa$. The speaker will prove a theorem of Julia Knight that there is a Jónsson $\omega_1$-like model of set theory, using a construction of Ali Enayat.

# Grounded Martin’s axiom

*I will present a weakening of Martin’s axiom which asserts the existence of partial generics only for ccc posets contained in a ccc ground model. This principle, named the grounded Martin’s axiom, emerges naturally in the analysis of the Solovay-Tennenbaum proof of the consistency of MA. While the grounded MA has some of the combinatorial consequences of MA, it will be shown to be more flexible (being consistent with a singular continuum, for example) and more robust under forcing (being preserved in a strong way under both adding a Cohen or a random real).*

# Some problems motivated by nonstandard set theory

Nonstandard set theory enriches the usual set theory by a unary “standardness” predicate. Investigations of its foundations raise a number of questions that can be formulated in ZFC or GB and appear open. I will discuss several such problems concerning elementary embeddings, ultraproducts, ultrafilters and large cardinals.

# Killing Inaccessible Cardinals Softly

I shall introduce the killing-them-softly phenomenon among large cardinals by showing how it works for inaccessible and Mahlo cardinals. A large cardinal is killed softly whenever, by forcing, one of its large cardinal properties is destroyed while as many as possible weaker large cardinal properties, below this one, are preserved. I shall also explore the various degrees of inaccessibility and show Mahlo cardinals are $alpha$-hyper$^{beta}$-inaccessible and beyond.

# Survey on the structure of the Tukey theory of ultrafilters

The Tukey order on ultrafilters is a weakening of the well-studied Rudin-Keisler order, and the exact relationship between them is a question of interest. In second vein, Isbell showed that there is a maximum Tukey type among ultrafilters and asked whether there are others. These two questions are the main guiding forces of the current research. In this talk, we present highlights of recent work of Blass, Dobrinen, Mijares, Milovich, Raghavan, Todorcevic, and Trujillo (in various combinations for various papers). Further information about results mentioned in this talk can be found in a recent survey article by the speaker.

# Canonical Ramsey theory on Polish spaces

I would like to give an overview of recent results in canonical Ramsey theory in the context of descriptive set theory. This is the subject of a recent monograph joint with with Vladimir Kanovei and Jindra Zapletal. The main question we address is the following. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? Canonical Ramsey theory stems from finite combinatorics and is concerned with finding canonical forms of equivalence relations on finite (or countable) sets. We obtain canonization results for analytic and Borel equivalence relations and in cases when canonization is impossible, we prove ergodicity theorems. For a publisher’s book description see:

# Embeddings among the $\omega_1$-like models of set theory, part II

The speaker will give the second part of her talk, continued from the previous week. An $\omega_1$-like model of set theory is an uncountable model, all of whose initial segments are countable. The speaker will present two $\omega_1$-like models of set theory, constructed using $\Diamond$, which are incomparable with respect to embeddability: neither is isomorphic to a submodel of the other. Under a suitable large cardinal assumption, there are such models that are well-founded.

# Embeddings among the $\omega_1$-like models of set theory, part I

An $\omega_1$-like model of set theory is an uncountable model, all of whose initial segments are countable. The speaker will present two $\omega_1$-like models of set theory, constructed using $\Diamond$, which are incomparable with respect to embeddability: neither is isomorphic to a submodel of the other. Under a suitable large cardinal assumption, there are such models that are well-founded.

# A self-specializing Souslin tree

I will present a construction, assuming Jensen’s combinatorial principle diamond, of a Souslin tree T which, after forcing with T, will be Ahronszajn off the generic branch. More precisely, forcing with T will add a cofinal branch b through T, yet in the generic extension by b, whenever p is a node of T which does not belong to b, then the subtree of T which lies above p will be Q-embeddable, meaning that there is an order preserving function from that subtree to the rationals. This shows that the rigidity property of being Souslin off the generic branch is strictly stronger than the unique branch property, two notions of rigidty previously studied in joint work with Joel Hamkins, where it was conjectured that it would be possible to construct such a self specializing Souslin tree.

# The role of the axiom of foundation in the Kunen inconsistency

The axiom of foundation plays an interesting role in the Kunen inconsistency, the assertion that there is no nontrivial elementary embedding of the set-theoretic universe to itself, for the truth or falsity of the Kunen assertion depends on one’s specific anti-foundational stance. The fact of the matter is that different anti-foundational theories come to different conclusions about this assertion. On the one hand, it is relatively consistent with ZFC without foundation that the Kunen assertion fails, for there are models of ZFC-F in which there are definable nontrivial elementary embeddings $j:Vto V$. Indeed, in Boffa’s anti-foundational theory BAFA, the Kunen assertion is outright refutable, and in this theory there are numerous nontrivial elementary embeddings of the universe to itself. Meanwhile, on the other hand, Aczel’s anti-foundational theory GBC-F+AFA, as well as Scott’s theory GBC-F+SAFA and other anti-foundational theories, continue to prove the Kunen assertion, ruling out the existence of a nontrivial elementary embedding $j:Vto V$.

This is very recent joint work with Emil Jeřábek, Ali Sadegh Daghighi and Mohammad Golshani, based on an interaction growing out of Ali’s question on MathOverflow. Our paper will be completed soon.

# Algebraicity and implicit definability in set theory

An element *a* is *definable* in a model M if it is the unique object in M satisfying some first-order property. It is *algebraic*, in contrast, if it is amongst at most finitely many objects satisfying some first-order property φ, that is, if { b | M satisfies φ[b] } is a finite set containing *a*. In this talk, I aim to consider the situation that arises when one replaces the use of definability in several parts of set theory with the weaker concept of algebraicity. For example, in place of the class HOD of all hereditarily ordinal-definable sets, I should like to consider the class HOA of all hereditarily ordinal algebraic sets. How do these two classes relate? In place of the study of pointwise definable models of set theory, I should like to consider the pointwise algebraic models of set theory. Are these the same? In place of the constructible universe L, I should like to consider the inner model arising from iterating the algebraic (or implicit) power set operation rather than the definable power set operation. The result is a highly interest new inner model of ZFC, denoted Imp, whose properties are only now coming to light. Is Imp the same as L? Is it absolute? I shall answer all these questions at the talk, but many others remain open.

This is joint work with Cole Leahy (MIT).

# Indestructibility for Ramsey cardinals

A large cardinal $\kappa$ is said to be indestructible by a certain poset $\mathbb P$ if $\kappa$ retains the large cardinal property in all forcing extensions by $\mathbb P$. Since most relative consistency results for ${\rm ZFC}$ are obtained via forcing, the knowledge of a large cardinal’s indestructibility properties is used to establish the consistency of that large cardinal with other set theoretic properties. In this talk, I will use an elementary embeddings characterization of Ramsey cardinals to prove some basic indestructibility results.

# An elementary embeddings characterization of Ramsey cardinals

A cardinal $\kappa$ is Ramsey if every coloring of the finite subsets of $\kappa$ in two colors has a homogeneous set of size $\kappa$. In this talk, I will motivate and prove some basics facts about the little known but very elegant and useful elementary embeddings characterization of Ramsey cardinals.

# Dissertation Defense: Inverse limits of models of set theory and the large cardinal hierarchy near a high-jump cardinal

This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other.

The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse-directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards fully characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. If the inverse limit exists, then it is given by either the entire thread class or a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved in both directions by forcing under fairly general assumptions. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter.

The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing. A high-jump cardinal is the critical point of an elementary embedding $j: V to M$ such that $M$ is closed under sequences of length $supset{j(f)(kappa) st f: kappa to kappa}$. Two of the most important results in the chapter are as follows. A Vopenka cardinal is equivalent to an Woodin-for-supercompactness cardinal. The existence of an excessively hypercompact cardinal is inconsistent.

# Ramsey ultrafilters

I will introduce the concept of a Ramsey ultrafilter and show that under Martin’s Axiom, and under the continuum hypothesis, Ramsey ultrafilters exists. I will actually show that this follows from some consequences of MA on cardinal invariants of the continuum. If time permits, I will make a connection to Ramsey’s theorem. This talk is intended to bridge the gap between the previous talk by Miha Habic on Martin’s Axiom and the upcoming talks by Victoria Gitman on Ramsey cardinals.

# Martin’s Axiom

Martin’s Axiom is the prototypical forcing axiom, asserting that partial generics exist for ccc forcing. I will present the classical proof of its consistency, due to Solovay and Tennenbaum. If time permits I will also describe some of the consequences of Martin’s Axiom, in particular its effect on the cardinal characteristics of the continuum.

# Forcing for Constructive Set Theory

As is well known, forcing is the same as Boolean-valued models. If instead of a Boolean algebra one used a Heyting algebra, the result is a Heyting-valued model. The result then typically models only constructive logic and falsifies Excluded Middle. On the one hand, many of the same intuitions from forcing carry over, while on the other the result is quite foreign to a classical mathematician. I will give a survey of perhaps too many examples, and call for the importation of more methods from current classical set-theory into constructivism.

# The omega one of chess

This talk will be based on my recent paper with C. D. A. Evans, Transfinite game values in infinite chess.

Infinite chess is chess played on an infinite chessboard. Since checkmate, when it occurs, does so after finitely many moves, this is technically what is known as an *open* game, and is therefore subject to the theory of open games, including the theory of ordinal game values. In this talk, I will give a general introduction to the theory of ordinal game values for ordinal games, before diving into several examples illustrating high transfinite game values in infinite chess. The supremum of these values is the *omega one of chess*, denoted by $omega_1^{mathfrak{Ch}}$ in the context of finite positions and by $omega_1^{mathfrak{Ch}_{hskip-2ex atopsim}}$ in the context of all positions, including those with infinitely many pieces. For lower bounds, we have specific positions with transfinite game values of $omega$, $omega^2$, $omega^2cdot k$ and $omega^3$. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, true $omega_1$.

# No seminar on February 26, the planned talk was rescheduled for March 5.

# Cardinals that might be viewed as miniature superstrong cardinals consistent with V=L

We will discuss cardinals that we may call superstrongly unfoldable cardinals. A cardinal kappa is superstrongly unfoldable if it is theta-superstrongly unfoldable for every ordinal theta, meaning that for subset A of kappa there is a nontrivial elementary embedding j:M–>N between transitive ZFC models having critical point kappa such that j(kappa)>theta and V_{j(kappa)} is a subset of N and the set A is an element of M. Superstrongly unfoldable cardinals lie in consistency strength between weakly compact cardinals and Ramsey cardinals, and they are relatively consistent with V=L. We will show the tighter consistency bounds of strongly unfoldable cardinals below and subtle cardinals above. This is joint work with Joel David Hamkins.

# Magidor Forcing and iterations

I am going to use the characterization of Magidor sequences I gave last Friday, in order to show that the critical sequences of certain iterations are Magidor generic over the limit model. I am also going to say a few words about maximality and uniqueness properties of Magidor sequences, which can be shown using the connection between Magidor forcing and iterations.

# Magidor Forcing

In this talk, I am going to present a forcing designed by Magidor in the late seventies to change the cofinality of a measurable cardinal without collapsing cardinals. Previously, Prikry had introduced a forcing that changes the cofinality of a measurable cardinal to $omega$. Magidor’s forcing has more flexibility, but needs stronger assumptions also, and it is quite complex. After giving some background and showing the basic properties of Magidor forcing, I will prove a combinatorial characterization of genericity of sequences added by the forcing. There are some similarities to the situation of Prikry forcing, where an $omega$-sequence of ordinals less than the measurable cardinal is generic iff it is almost contained in any measure one set, as was shown by Mathias. I will show or sketch a proof of the corresponding characterization in the case of Magidor forcing in the second part of the talk next week.

# Superstrong cardinals are never Laver indestructible, and neither are extendible, almost huge and rank-into-rank cardinals

Although the large cardinal indestructibility phenomenon, initiated with Laver’s seminal 1978 result that any supercompact cardinal $kappa$ can be made indestructible by $ltkappa$-directed closed forcing and continued with the Gitik-Shelah treatment of strong cardinals, is by now nearly pervasive in set theory, nevertheless I shall show that no superstrong cardinal—and hence also no $1$-extendible cardinal, no almost huge cardinal and no rank-into-rank cardinal—can be made indestructible, even by comparatively mild forcing: all such cardinals $kappa$ are destroyed by $Add(kappa,1)$, by $Add(kappa,kappa^+)$, by $Add(kappa^+,1)$ and by many other commonly considered forcing notions.

This is very recent joint work with Konstantinos Tsaprounis and Joan Bagaria.

# The Laver Preparation

I shall show how to make a supercompact cardinal kappa indestructible for

# The strength of the failure of the Kurepa Hypothesis

I will show that Diamond Plus holds in inner models of the form L[A], for subsets A of aleph one in the sense of L[A]. Putting this together with the result from last meeting, that Diamond Plus implies the Kurepa Hypothesis, I will show that if the Kurepa Hypothesis fails, then aleph two is an inaccessible cardinal in L. Again, putting this together with another result from the previous seminar meeting, that one can force the failure of Kurepa’s Hypothesis over a model with an inaccessible cardinal, this shows the equiconsistency of the failure of Kurepa’s Hypothesis with an inaccessible cardinal, over ZFC. These results are mainly due to Silver and Solovay.

# Diamond Plus and the Kurepa Hypothesis

The speaker will prove some connections between Diamond Plus and the Kurepa Hypothesis.

# Extending a property of HOD-supercompactness. Or Not.

I will discuss a result of Sargsyan and his method of proof in order to show why a theorem from my dissertation was incorrect, and some of the interesting results we discovered in an effort to save the theorem.

# Preservation of DC delta by forcing with a closure point at delta

Richard Laver [2007] showed that if M satisfies ZFC and G is any M-generic filter for forcing P of size less than delta, then M is definable in M[G] from parameter P(delta)^M. I will discuss a generalization of this result for models M that satisfy ZF but only a small fragment of the axiom of choice. This is joint work with Victoria Gitman.

Definition (ZF). P*Q has closure point delta if P is well-orderable of size at most delta and Q is <=delta strategically closed. (Q need not be well-orderable here.) Theorem: If M models ZF+DC_delta and P is forcing with closure point delta, then M is definable in M[G] from parameter P(delta)^M.

# Definability of the ground model in forcing extensions of ZF-models

Richard Laver [2007] showed that if M satisfies ZFC and G is any M-generic filter for forcing P of size less than delta, then M is definable in M[G] from parameter P(delta)^M. I will discuss a generalization of this result for models M that satisfy ZF but only a small fragment of the axiom of choice. This is joint work with Victoria Gitman.

Definition (ZF). P*Q has closure point delta if P is well-orderable of size at most delta and Q is <=delta strategically closed. (Q need not be well-orderable here.) Theorem: If M models ZF+DC_delta and P is forcing with closure point delta, then M is definable in M[G] from parameter P(delta)^M.

# Generalized Solovay Measures

We will introduce some background and recent progress made in solving the following open problem:

Determine the exact consistency strength of the theory T = ZF + DC + $omega_1$ is supercompact.

It’s known that the upper-bound consistency strength for T is a class of Woodin limits of Woodin cardinals which is (surprisingly) much weaker than ZFC + a supercompact. We will discuss how one might go about computing lower-bounds for T. If time allows, we’ll briefly talk about the relationship of T with the Chang model (CM) and its generalization (CM^+).

# Generic embeddings from indestructible weak compactness, II

I will give an introduction to and some applications of a type of generic embeddings that arises under the assumption of an indestructible weakly compact cardinal.

# Generic embeddings from indestructible weak compactness, I

I will give an introduction to and some applications of a type of generic embeddings that arises under the assumption of an indestructible weakly compact cardinal.

# The least weakly compact cardinal can be unfoldable, weakly measurable and nearly theta-supercompact

Starting from suitable large cardinal hypothesis, I will explain how to force the least weakly compact cardinal to be unfoldable, weakly measurable and, indeed, nearly θ-supercompact. These results, proved in joint work with Jason Schanker, Moti Gitik and Brent Cody, exhibit an identity-crises phenomenon for weak compactness, similar to the phenomenon identified by Magidor for the case of strongly compact cardinals.

# The equivalence of Woodinized supercompact cardinals and Vopenka cardinals

I present a tentative result that Woodin for supercompactness cardinals are equivalent to Vopenka cardinals. This result is vaguely hinted at, though not proven, in Kanamori’s text, and I believe I have worked out the details. Kappa is *Vopenka* iff for every collection of kappa many model-theoretic structures with domain subset of $V_kappa$ there exists an elementary embedding between two of them. Kappa is Woodin for supercompactness if it meets the definition of a Woodin cardinal, with strongness replaced by supercompactness. That is to say, for every function $f:kappatokappa$, there exists a closure point delta of f and an elementary embedding $j:Vto M$ such that $j(delta)ltkappa$ and $latex M$ is closed in $latex V$ under $j(f)(delta)$ sequences.

# The omega one of infinite chess

Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-$n$ problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most $n$ moves. A naive formulation of this problem leads to assertions of high arithmetic complexity with $2n$ alternating quantifiers—there is a move for white, such that for every black reply, there is a countermove for white, and so on. In such a formulation, the problem does not appear to be decidable; and one cannot expect to search an infinitely branching game tree even to finite depth. Nevertheless, in joint work with Dan Brumleve and Philipp Schlicht, confirming a conjecture of myself and C. D. A. Evans, we establish that the mate-in-$n$ problem of infinite chess is computably decidable, uniformly in the position and in $n$. Furthermore, there is a computable strategy for optimal play from such mate-in-$n$ positions. The proof proceeds by showing that the mate-in-$n$ problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. An equivalent account of the result arises from the realization that the structure of chess is interpretable in the standard model of Presburger arithmetic $langlemathbb{N},+rangle$. Unfortunately, this resolution of the mate-in-$n$ problem does not appear to settle the decidability of the more general winning-position problem, the problem of determining whether a designated player has a winning strategy from a given position, since a position may admit a winning strategy without any bound on the number of moves required. This issue is connected with transfinite game values in infinite chess, and the exact value of the omega one of chess $omega_1^{rm chess}$ is not known. I will also discuss recent joint work with C. D. A. Evans and W. Hugh Woodin showing that the omega one of infinite positions in three-dimensional infinite chess is true $omega_1$: every countable ordinal is realized as the game value of such a position.