Set Theory Seminar

Any ultrapower $M$ of the universe by a normal measure on a cardinal $\kappa$ is quite far from $V$ in the sense that it computes $V_{\kappa+2}$ incorrectly. If GCH holds, this amounts to saying that $M$ is missing a subset of $\kappa^+$. Steel asked whether, even in the absence of GCH, normal ultrapowers at $\kappa$ must miss a subset of $\kappa^+$. In the early 90s Cummings gave a negative answer, building a model with a normal measure on $\kappa$ whose ultrapower captures the entire powerset of $\kappa^+$. I will present some joint work with Radek Honzík in which we improved Cummings’ result to get this capturing property to hold at the least measurable cardinal.

Video

Let $\Phi$ be the statement that for any function $f: \omega_1 \times \omega_1 \to \omega$, there are functions $g_1, g_2 : \omega_1 \to \omega$ such that for all $(x,y) \in \omega_1 \times \omega_1$, we have $f(x,y) \le \text{max }\{g_1(x), g_2(y)\}$. We will show that $\Phi$ follows from ${\rm ZF} + {\rm DC} + `\omega_1\text{ is measurable'}$. On the other hand using core models, we will show that $\Phi + `\text{the club filter on }\omega_1\text{ is normal'}$ implies there are inner models with many measurable cardinals. We conjecture that $\Phi$ and ${\rm ZF} + {\rm DC} + `\omega_1\text{ is measurable'}$ have the same consistency strength. The research is joint with Francois Dorais at the University of Vermont.

Video

I have been working over the past few years on the project of trying to improve our understanding of the forcing axiom for subcomplete forcing. The most compelling feature of this axiom is its consistency with the continuum hypothesis. On the other hand, it captures many of the major consequences of Martin's Maximum. It is a compelling feature of Martin's Maximum that many of its consequences filter through Todorcevic's Strong Reflection Principle SRP. SRP has some consequences that the subcomplete forcing axiom does not have, like the failure of CH and the saturation of the nonstationary ideal. It has been unclear until recently whether there is a version of SRP that relates to the subcomplete forcing axiom as the full SRP relates to Martin's Maximum, but it turned out that there is: I will detail how to associate in a canonical way to an arbitrary forcing class its corresponding fragment of SRP in such a way that (1) the forcing axiom for the forcing class implies its fragment of SRP, (2) the stationary set preserving fragment of SRP is the full principle SRP, and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. I will describe how this association works, describe some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity, and say a little more about the extent of (3).

Video

To what extent can formulas from infinitary logics be used in set-theoretic reflection arguments? If $\kappa$ is a measurable cardinal, any $L_{\kappa,\kappa}$ sentence which is true in $(\kappa,\in)$, must be true about some strictly smaller cardinal. Whereas, there are $L_{\kappa^+,\kappa^+}$ sentences of length $\kappa$ which are true in $(\kappa,\in)$ and which are not true about any smaller cardinal. However, if $\kappa$ is a measurable cardinal and some $L_{\kappa^+,\kappa^+}$ sentence $\varphi$ is true in $(\kappa,\in)$, then there must be some strictly smaller cardinal $\alpha<\kappa$ such that a canonically restricted version of $\varphi$ holds about $\alpha$. Building on work of Bagaria and Sharpe-Welch, we use canonical restriction of formulas to define notions of $\Pi^1_\xi$-indescribability of a cardinal $\kappa$ for all $\xi<\kappa^+$. In this context we show that such higher indescribability hypotheses are strictly weaker than measurability, we prove the existence of universal $\Pi^1_\xi$-formulas, study the associated normal ideals and notions of $\xi$-clubs and prove a hierarchy result. Time permitting we will discuss some applications.

Video

This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions from $\omega^\omega$ to $\omega^\omega$. I prove that these cardinals can be organized into two diagrams analogous to the standard Cichoń diagram show several independence results and investigate their relation to cardinal invariants on omega. In the second half of the thesis I look at forcing axioms compatible with CH. First I consider Jensen's subcomplete and subproper forcing. I generalize these notions to larger classes which are (apparently) much more nicely behaved structurally. I prove iteration and preservation theorems for both classes and use these to produce many new models of the subcomplete forcing axiom. Finally I deal with dee-complete forcing and its associated axiom DCFA. Extending a well-known result of Shelah, I show that if a tree of height $\omega_1$ with no branch can be embedded into an $\omega_1$ tree, possibly with uncountable branches, then it can be specialized without adding reals. As a consequence I show that DCFA implies there are no Kurepa trees, even if CH fails.

An old result of Isbell characterises measurable cardinals in terms of certain canonical limits in the category of sets. After introducing this characterisation, I will talk about recent work with Adamek, Campion, Positselski and Rosicky teasing out the importance of the canonicity for this and related results. The language will be category-theoretic but the proofs will be quite hands-on combinatorial constructions with sets.

The maximality principle (${\rm MP}$) is the assertion that any sentence which can be forced in such a way that after any further forcing the sentence remains true, must already be true. In modal terms, ${\rm MP}$ states that forceably necessary sentences are true. The resurrection axiom (${\rm RA}$) asserts that the ground model is as existentially closed in its forcing extensions as possible. In particular, ${\rm RA}$ relative to $H_{\mathfrak c}$ states that for every forcing $\mathbb Q$ there is a further forcing $\mathbb R$ such that $H_{\mathfrak c}^V \prec H_{\mathfrak c}^{V[G][H]}$, for $G*H \subseteq \mathbb Q *\dot{\mathbb R}$ generic.

It is reasonable to ask whether ${\rm MP}$ and ${\rm RA}$ can consistently both hold. I showed that indeed they can, and that ${\rm RA}+{\rm MP}$ is equiconsistent with a strongly uplifting fully reflecting cardinal, which is a combination of the large cardinals used to force the principles separately. In this talk I give a sketch of the equiconsistency result.

Many prominent large cardinal notions up to measurability can be characterized by the existence of certain ultrafilters for small models of set theory. Most prominently, this includes weakly compact, ineffable, Ramsey and completely ineffable cardinals, but there are many more, and our characterization schemes also give rise to many new natural large cardinal concepts. Moreover, these characterizations allow for the uniform definition of ideals associated to these large cardinals, which agree with the ideals from the set-theoretic literature (for example, the weakly compact, the ineffable, the Ramsey or the completely ineffable ideal) whenever such had been previously established. For many large cardinal notions, we can show that their ordering with respect to direct implication, but also with respect to consistency strength corresponds in a very canonical way to certain relations between their corresponding large cardinal ideals. This is all material from a fairly extensive joint paper with Philipp Luecke, and I will try to provide an overview as well as present some particular results from this paper.

We will discuss some recent ZFC results concerning the generalized Baire spaces, and more specifically the generalized bounding number, relatives of the generalized almost disjointness number, as well as generalized reaping and domination.

Zermelo famously characterized the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$ in his 1930 quasi-categoricity result asserting that the models of $\text{ZFC}_2$ are precisely those isomorphic to a rank-initial segment $V_\kappa$ of the cumulative set-theoretic universe $V$ cut off at an inaccessible cardinal $\kappa.$ I shall discuss the extent to which Zermelo's quasi-categoricity analysis can rise fully to the level of categoricity, in light of the observation that many of the $V_\kappa$ universes are categorically characterized by their sentences or theories. For example, if $\kappa$ is the smallest inaccessible cardinal, then up to isomorphism $V_\kappa$ is the unique model of $\text{ZFC}_2$ plus the sentence 'there are no inaccessible cardinals.' This cardinal $\kappa$ is therefore an instance of what we call a first-order sententially categorical cardinal. Similarly, many of the other inaccessible universes satisfy categorical extensions of $\text{ZFC}_2$ by a sentence or theory, either in first or second order. I shall thus introduce and investigate the categorical cardinals, a new kind of large cardinal. This is joint work with Robin Solberg (Oxford).

Video

The notion of a guessing model introduced by Viale and Weiss. The principle ${\rm GM}(\omega_2,\omega_1)$ asserts that there are stationary many guessing models of size $\aleph_1$ in $H_\theta$, for all large enough regular $\theta$. It follows from ${\rm PFA}$ and implies many of its structural consequences, however it does not settle the value of the continuum. In search of higher of forcing axioms it is therefore natural to look for extensions and higher versions of this principle. We formulate and prove the consistency of one such statement that we call ${\rm SGM}^+(\omega_3,\omega_1)$.
It has a number of important structural consequences:

- the tree property at $\aleph_2$ and $\aleph_3$
- the failure of various weak square principles
- the Singular Cardinal Hypothesis
- Mitchell’s Principle: the approachability ideal agrees with the non stationary ideal on the set of $\text{cof}(\omega_1)$ ordinals in $\omega_2$
- Souslin’s Hypothesis
- The negation of the weak Kurepa Hypothesis
- Abraham’s Principles: every forcing which adds a subset of $\omega_2$ either adds a real or collapses some cardinals, etc.

The results are joint with my PhD students Rahman Mohammadpour.

Slides

Recursive saturation, introduced by J. Barwise and J. Schlipf is a robust notion which has proved to be important for the study of nonstandard models (in particular, it is ubiquitous in the model theory of axiomatic theories of truth, e.g. in the topic of satisfaction classes, where one can show that if $M \models ZFC$ is a countable $\omega$-nonstandard model, then $M$ admits a satisfaction class iff $M$ is recursively saturated). V. Gitman and J. Hamkins showed in A Natural Model of the Multiverse Axioms that the collection of countable, recursively saturated models of set theory satisfy the so-called Hamkins's Multiverse Axioms. The property that forces all the models in the Multiverse to be recursively saturated is the so-called Well-Foundedness Mirage axiom which asserts that every universe is $\omega$-nonstandard from the perspective of some larger universe, or to be more precise, that: if a model $M$ is in the multiverse then there is a model $N$ in the multiverse such that $M$ is a set in $N$ and $N \models 'M \text{ is }\omega-\text{nonstandard.'}$. Inspection of the proof led to a question if the recursive saturation could be avoided in the Multiverse by weakening the Well-Foundedness Mirage axiom. Our main results answer this in the positive. We give two different versions of the Well-Foundedness Mirage axiom - what we call Weak Well-Foundedness Mirage (saying that if $M$ is a model in the Multiverse then there is a model $N$ in the Multiverse such that $M \in N$ and $N \models 'M \text{ is nonstandard.'}$.) and Covering Well-Foundedness Mirage (saying that if $M$ is a model in the Multiverse then there is a model $N$ in the Multiverse with $K \in N$ such that $K$ is an end-extension of $M$ and $N \models 'K \text{ is } \omega-\text{nonstandard.'}$). I will present constructions of two different Multiverses satisfying these two weakened axioms. This is joint work with V. Gitman. T. Meadows and K. Williams.

Recursive saturation, introduced by J. Barwise and J. Schlipf is a robust notion which has proved to be important for the study of nonstandard models (in particular, it is ubiquitous in the model theory of axiomatic theories of truth, e.g. in the topic of satisfaction classes, where one can show that if $M \models ZFC$ is a countable $\omega$-nonstandard model, then $M$ admits a satisfaction class iff $M$ is recursively saturated). V. Gitman and J. Hamkins showed in A Natural Model of the Multiverse Axioms that the collection of countable, recursively saturated models of set theory satisfy the so-called Hamkins's Multiverse Axioms. The property that forces all the models in the Multiverse to be recursively saturated is the so-called Well-Foundedness Mirage axiom which asserts that every universe is $\omega$-nonstandard from the perspective of some larger universe, or to be more precise, that: if a model $M$ is in the multiverse then there is a model $N$ in the multiverse such that $M$ is a set in $N$ and $N \models 'M \text{ is }\omega-\text{nonstandard.'}$. Inspection of the proof led to a question if the recursive saturation could be avoided in the Multiverse by weakening the Well-Foundedness Mirage axiom. Our main results answer this in the positive. We give two different versions of the Well-Foundedness Mirage axiom - what we call Weak Well-Foundedness Mirage (saying that if $M$ is a model in the Multiverse then there is a model $N$ in the Multiverse such that $M \in N$ and $N \models 'M \text{ is nonstandard.'}$.) and Covering Well-Foundedness Mirage (saying that if $M$ is a model in the Multiverse then there is a model $N$ in the Multiverse with $K \in N$ such that $K$ is an end-extension of $M$ and $N \models 'K \text{ is } \omega-\text{nonstandard.'}$). I will present constructions of two different Multiverses satisfying these two weakened axioms. This is joint work with V. Gitman. T. Meadows and K. Williams.

An inner model is a ground if V is a set forcing extension of it. The intersection of the grounds is the mantle, an inner model of ZFC which enjoys many nice properties. Fuchs, Hamkins, and Reitz showed that the mantle is highly malleable. Namely, they showed that every model of set theory is the mantle of a bigger, better universe of sets. This then raises the possibility of iterating the definition of the mantle—the mantle, the mantle of the mantle, and so on, taking intersections at limit stages—to obtain even deeper inner models. Let's call the inner models in this sequence the inner mantles.

In this talk I will present some results about the sequence of inner mantles, answering some questions of Fuchs, Hamkins, and Reitz. Specifically, I will present the following results, analogues of classic results about the sequence of iterated HODs.

1. (Joint with Reitz) Consider a model of set theory and consider an ordinal eta in that model. Then this model has a class forcing extension whose eta-th inner mantle is the model we started out with, where the sequence of inner mantles does not stabilize before eta.

2. It is consistent that the omega-th inner mantle is an inner model of ZF + ¬AC.

3. It is consistent that the omega-th inner mantle is not a definable class, and indeed fails to satisfy Collection.

A model $M$ of set theory is said to be 'condensable' if there is an 'ordinal' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is both isomorphic to $M$, and an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M$. Clearly if $M$ is condensable, then $M$ is ill-founded. The work of Barwise and Schlipf in the mid 1970s showed that countable recursively saturated models of ZF are condensable.

In this two-part talk, we present a number of new results related to condensable models, including the following two theorems.

Theorem 1. Assuming that there is a well-founded model of ZFC plus 'there is an inaccessible cardinal', there is a condensable model $M$ of ZFC which has the property that every definable element of $M$ is in the well-founded part of $M$ (in particular, $M$ is $\omega$-standard, and therefore not recursively saturated).

Theorem 2. The following are equivalent for an ill-founded model $M$ of ZF of any cardinality:
(a) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension.
(b) There is a cofinal subset of 'ordinals' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M.$
Moreover, if $M$ is a countable ill-founded model of ZFC, then conditions (a) and (b) above are equivalent to:
(c) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension + $\Sigma^1_1$-Choice.

Slides

A model $M$ of set theory is said to be 'condensable' if there is an 'ordinal' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is both isomorphic to $M$, and an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M$. Clearly if $M$ is condensable, then $M$ is ill-founded. The work of Barwise and Schlipf in the mid 1970s showed that countable recursively saturated models of ZF are condensable.

In this two-part talk, we present a number of new results related to condensable models, including the following two theorems.

Theorem 1. Assuming that there is a well-founded model of ZFC plus 'there is an inaccessible cardinal', there is a condensable model $M$ of ZFC which has the property that every definable element of $M$ is in the well-founded part of $M$ (in particular, $M$ is $\omega$-standard, and therefore not recursively saturated).

Theorem 2. The following are equivalent for an ill-founded model $M$ of ZF of any cardinality:
(a) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension.
(b) There is a cofinal subset of 'ordinals' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M.$
Moreover, if $M$ is a countable ill-founded model of ZFC, then conditions (a) and (b) above are equivalent to:
(c) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension + $\Sigma^1_1$-Choice.

Slides

Computing the large cardinal strength of a given statement is one of the key research directions in set theory. Fruitful tools to tackle such questions are given by inner model theory. The study of inner models was initiated by Gödel's analysis of the constructible universe $L$. Later, it was extended to canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others.

We will outline two recent applications where inner model theory is used to obtain lower bounds in large cardinal strength for statements that do not involve inner models. The first result, in part joint with J. Aguilera, is an analysis of the strength of determinacy for certain infinite two player games of fixed countable length, and the second result, joint with Y. Hayut, involves combinatorics of infinite trees and the perfect subtree property for weakly compact cardinals $\kappa$. Finally, we will comment on obstacles, questions, and conjectures for lifting these results higher up in the large cardinal hierarchy.

Slides

In May 2019 Trevor Wilson proved that the Weak Vopenka Principle (WVP), which asserts that the opposite of the category of Ordinals cannot be fully embedded into the category of Graphs, is equivalent to the class of ordinals being Woodin. In particular this implies that WVP is not equivalent to Vopenka’s Principle, thus solving an important long-standing open question in category theory. I will report on a joint ensuing work with Trevor Wilson in which we analyse the strength of WVP for definable classes of full subcategories of Graphs, obtaining exact level-by-level characterisations in terms of a natural hierarchy of strong cardinals.

Slides

Starting from a model of ZFC with two supercompact cardinals, I will discuss how to force and construct a model in which the first two strongly compact cardinals $\kappa_1$ and $\kappa_2$ are also the first two measurable cardinals. In this model, $\kappa_1$'s strong compactness is indestructible under arbitrary $\kappa_1$-directed closed forcing, and $\kappa_2$'s strong compactness is indestructible under ${\rm Add}(\kappa_2, \lambda)$ for any ordinal $\lambda$. This answers a generalized version of a question of Sargsyan.

Slides

An important advancement in iterated forcing was Jensen’s proof that CH does not imply $\diamondsuit$ by iteratively specializing Aronszajn trees with countable levels without adding reals thus producing a model of CH plus 'all Aronszajn trees are special'. This proof was improved by Shelah who developed a general method around the notion of dee-complete forcing. This class (under certain circumstances) can be iterated with countable support and does not add reals. However, neither Jensen's nor Shelah's posets will specialize trees of uncountable width and it remains unclear when one can iteratively specialize wider trees. Indeed a very intriguing example, due to Todorčević, shows that there is always a wide Aronszajn tree which cannot be specialized without adding reals. By contrast the ccc forcing for specializing Aronszajn trees makes no distinction between trees of different widths (but may add many reals). In this talk we will provide a general criteria a wide trees Aronszajn tree can have that implies the existence of a dee-complete poset specializing it. Time permitting we will discuss applications of this forcing to forcing axioms compatible with CH and some open questions related to set theory of the reals.

Slides

Issues of set theoretic compactness frequently arise when considering questions from homological algebra about derived functors. In particular, the non-vanishing of such derived functors is often witnessed by a concrete combinatorial instance of set theoretic incompactness, so that homological questions can be translated into questions about combinatorial set theory. In this talk, we will discuss some recent results about the derived functors of the inverse limit functor. We will focus on a specific inverse system of abelian groups, $\mathbf{A}$, that arose in Mardešić and Prasolov's work on the additivity of strong homology and has since arisen independently in a number of contexts. Our main result states that, relative to the consistency of a weakly compact cardinal, it is consistent that the $n$-th derived limits $\lim^n \mathbf{A}$ vanish simultaneously for all $n \geq 1$. We will sketch a proof of this theorem and then discuss the extent to which certain generalizations of the result can hold. The arguments will be purely set theoretic, and no knowledge of homological algebra will be assumed. This is joint work with Jeffrey Bergfalk.