Set Theory Seminar

In these two talks, I will explain an untyped framework for the multiverse of set theory, developed in a joint paper with Graham Leigh. ZF is extended with semantically motivated axioms utilizing the new symbols Uni(U) and Mod(U, sigma), expressing that U is a universe and that sigma is true in the universe U, respectively. Here sigma ranges over the augmented language, leading to liar-style phenomena.

The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle, to the effect that the background theory of the multiverse does not hold a privileged position over the theories of its internal universes.

Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. This is applied to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory ZF, and thus not seriously limiting to the diversity of the set-theoretic multiverse. I also plan to discuss connections with Hamkins's multiverse theory, and the model of this constructed by Gitman and Hamkins. Throughout the talks I'm keen to discuss both philosophical and mathematical matters with the audience, concerning our Copernican approach to the multiverse of sets.

Video

In these two talks, I will explain an untyped framework for the multiverse of set theory, developed in a joint paper with Graham Leigh. ZF is extended with semantically motivated axioms utilizing the new symbols Uni(U) and Mod(U, sigma), expressing that U is a universe and that sigma is true in the universe U, respectively. Here sigma ranges over the augmented language, leading to liar-style phenomena.

The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle, to the effect that the background theory of the multiverse does not hold a privileged position over the theories of its internal universes.

Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. This is applied to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory ZF, and thus not seriously limiting to the diversity of the set-theoretic multiverse. I also plan to discuss connections with Hamkins's multiverse theory, and the model of this constructed by Gitman and Hamkins. Throughout the talks I'm keen to discuss both philosophical and mathematical matters with the audience, concerning our Copernican approach to the multiverse of sets.

Video

Video

Video

Video

Maximal almost disjoint (MAD) families and their relatives have been an important area of combinatorial and descriptive set theory since at least the 60s. In this talk I will discuss some relatives of MAD families, focussing on eventually different families of functions $f:\omega \to \omega$ and eventually different sets of permutations $p \in S(\omega)$. In the context of MAD families it has been fruitful to consider various strengthenings of the maximality condition to obtain several flavors of 'strongly' MAD families. One such strengthening that has proved useful in recent literature is that of tightness. Tight MAD families are Cohen indestructible and come with a properness preservation theorem making them nice to work with in iterated forcing contexts.

I will introduce a version of tightness for maximal eventually different families of functions $f:\omega \to \omega$ and maximal eventually different families of permutations $p \in S(\omega)$ respectively. These tight eventually different families share a lot of the nice, forcing theoretic properties of tight MAD families. Using them, I will construct explicit witnesses to $\mathfrak{a}_e= \mathfrak{a}_p = \aleph_1$ in many known models of set theory where this equality was either not known or only known by less constructive means. Working over $L$ we can moreover have the witnesses be $\Pi^1_1$ which is optimal for objects of size $\aleph_1$ in models where ${\rm CH}$ fails. These results simultaneously strengthen several known results on the existence of definable maximal sets of reals which are indestructible for various definable forcing notions. This is joint work with Vera Fischer.

Video

This oral exam talk will present a proof of Woodin's result that every real number is generic over some iterated ultrapower of any model with a Woodin cardinal. No fine structure theory will be used, and there will be a brief introduction to iteration trees.

Video

We define a set of real numbers to be perfectly small if it has perfectly many disjoint translates. Such sets have a strong intuitive claim to being probabilistically negligible, yet no non-trivial measure assigns them all a value of 0. We will prove from a moderate amount of choice that any total extension of Lebesgue measure concentrates on a perfectly small set, suggesting that for any such measure, translation-invariance fails 'as badly as possible.' From the ideas of this proof, we will also derive analogues of well-known paradoxes of randomness, specifically Freiling's symmetry paradox and the infinite prisoner hat puzzle, in terms of perfectly small sets. Finally, we discuss how these results constrain what a paradox-free set theory can look like and some related open questions.

A note on paradoxes

Video

In the past couple of years I have been involved (joint work with Väänänen and independently with Shelah) with some logics in the vicinity of Shelah's $\mathbb L^1_\kappa$ (a logic from 2012 that has Interpolation and a very weak notion of compactness, namely Strong Undefinability of Well-Orderings, and in some cases has a Lindström-type theorem for those two properties). Our work with Väänänen weakens the logic but keeps several properties. Our work with Shelah explores the connection with definability of AECs.

These logics seem to have additional interesting properties under the further assumption of strong compactness of a cardinal, and this brings them close to recent work of Boney, Dimopoulos, Gitman and Magidor [BDGM].

During the first lecture, I plan to describe two games and a syntax of two logics: Shelah's $\mathbb L^1_\kappa$ and my own logic (joint work with Väänänen) $\mathbb L^{1,c}_\kappa$. I will stress some of the properties of these logics, without any use of large cardinal assumptions.

During the second lecture, I plan to enter rather uncharted territory. I will describe some constructions done by Shelah (mostly) under the assumption of strong compactness, but I also plan to bring these logics to a territory closer to the work of [BDGM]. This second lecture will have more conjectures, ideas, and (hopefully interesting) discussions with some of the authors of that paper.

Video

In the past couple of years I have been involved (joint work with Väänänen and independently with Shelah) with some logics in the vicinity of Shelah's $\mathbb L^1_\kappa$ (a logic from 2012 that has Interpolation and a very weak notion of compactness, namely Strong Undefinability of Well-Orderings, and in some cases has a Lindström-type theorem for those two properties). Our work with Väänänen weakens the logic but keeps several properties. Our work with Shelah explores the connection with definability of AECs.

These logics seem to have additional interesting properties under the further assumption of strong compactness of a cardinal, and this brings them close to recent work of Boney, Dimopoulos, Gitman and Magidor [BDGM].

During the first lecture, I plan to describe two games and a syntax of two logics: Shelah's $\mathbb L^1_\kappa$ and my own logic (joint work with Väänänen) $\mathbb L^{1,c}_\kappa$. I will stress some of the properties of these logics, without any use of large cardinal assumptions.

During the second lecture, I plan to enter rather uncharted territory. I will describe some constructions done by Shelah (mostly) under the assumption of strong compactness, but I also plan to bring these logics to a territory closer to the work of [BDGM]. This second lecture will have more conjectures, ideas, and (hopefully interesting) discussions with some of the authors of that paper.

Video

The large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is known to be much stronger than the Axiom of Determinacy itself. In fact, Sargsyan conjectured it to be as strong as the existence of a cardinal that is both a limit of Woodin cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson showed that this would be optimal via a generalization of Woodin’s derived model construction. We will discuss a new translation procedure for hybrid mice extending work of Steel, Zhu and Sargsyan and use this to prove Sargsyan’s conjecture.

Video

The approximation property was introduced by Hamkins for his Gap Forcing Theorem, and it has turned out to be a very useful notion, appearing for example in the partial equiconsistency result of Viale and Weiss on PFA, and in the proof of Woodin's HOD Dichotomy Theorem. In the context of generic embeddings, there can be a useful interplay between elementarity and approximation. We discuss some recent work in this direction: (1) tensions between saturated ideals on $\omega_2$ and the tree property (with Sean Cox), (2) fragility of the strong independence spectra (with Vera Fischer), and (3) mutual inconsistency of Foreman‘s minimal generic hugeness axioms.

Video

The distinction between 'algebraic' and 'non-algebraic fields in mathematics, coined by Shapiro (1997), plays an important role in discussions about the status of set theory and connects back to the so-called universe/multiverse debate in the philosophy of set theory. In this talk we will see, that this distinction is not as clear cut as is usually assume when using it in the debate. In particular, we will see that in more recent formulations of this distinction, multiversism seems to split into a a strong and a weaker form. This can be translated to a meta-level, when considering the background theory in which set-theoretic multiversism can take place. This offers a more fine-grained picture of multiversism and allows us to mitigate a standard universist objection based on the conception of a multiversist background theory.

Video

Broad Infinity is a new and arguably intuitive axiom scheme in set theory. It states that 'broad numbers', which are three-dimensional trees whose growth is controlled, form a set. If the Axiom of Choice is assumed, then Broad Infinity is equivalent to the Ord-is-Mahlo scheme: every closed unbounded class of ordinals contains a regular ordinal.

Whereas the axiom of Infinity leads to generation principles for sets and families and ordinals, Broad Infinity leads to more advanced versions of these principles. The talk explains these principles and how they are related under various prior assumptions: the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.

Slides

Video

Assume $\mathcal{C}$ is the class of all linear orders $L$ such that $L$ is not a countable union of well ordered sets, and every uncountable subset of $L$ contains a copy of $\omega_1$. We show it is consistent that $\mathcal{C}$ has minimal elements. This answers an old question due to Galvin.

Slides

Video

The stationary reflection principle, which is often called the Weak Reflection Principle, is known to have many interesting consequences. As for cardinal arithmetic, it implies that $\lambda^\omega = \lambda$ for all regular cardinal $\lambda \geq \omega_2$. In this talk, we will discuss higher analogues of this stationary reflection principle and their consequences on cardinal arithmetic.

Video

Work in ZF and let $j:V_\alpha\to V_\alpha$ be an elementary, or partially elementary, embedding. One can examine the degree of uniqueness, definability or constructibility of $j$. For example, is there $\beta<\alpha$ such that $j$ is the unique (partially) elementary extension of $j\upharpoonright V_\beta$? Is $j$ definable from parameters over $V_\alpha$? We will discuss some results along these lines, illustrating that answers can depend heavily on circumstances. Some of the work is due independently and earlier to Gabriel Goldberg.

Video

Motivated by a classical theorem of Magidor, I will present results providing characterizations of important objects from the lower end of the large cardinal hierarchy through the existence of elementary embeddings between set-sized models that map their critical point to the large cardinal in question. Focusing on the characterization of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, I will show how these results can be used in the study of the combinatorics of strong chain conditions and the investigation of principles of structural reflection formulated by Bagaria.

Video

Indestructibility results of large cardinals have been an area of interest since Laver's 1978 proof that the supercompactness of $\kappa$ may be made indestructible by any $<\kappa$-directed closed forcing. I will present a continuation of this work, showing that $\alpha$-subcompact cardinals may be made suitably indestructible, but that for C(n)-supercompact cardinals this is largely not possible.

Video

Tarski proved (but didn't publish) the theorem that choice from pairs implies choice from four-element sets. Mostowski (1937) began a systematic study of such implications between choice axioms for families of finite sets. Gauntt (1970) completed that study (but didn't publish the results), obtaining equivalent characterizations in terms of fixed points of permutation groups. Truss (1973) extended Gauntt's results (and published this work).

It turns out that these finite choice axioms and their group-theoretic characterizations are instances of the same topos-theoretic statements, interpreted in two very different classes of topoi. My main result is an extension of that observation to the class of all topoi.

Most of my talk will be explaining the background: finite choice axioms, permutation groups, and a little bit about topoi - just enough to make sense of the main result. If time permits, I'll describe some of the ingredients of the proof.

Video

This talk is about the ongoing investigation of the relationships between measurable and strongly compact cardinals. I will present some of the history of the theorems in this theme, including Magidor's identity crisis, and give new results. The theorems presented are in particular about the relationships between strongly compact cardinals and measurable cardinals of different Mitchell orders. One of the main theorems is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, and where $\kappa_1$ is least with Mitchell order 1, and $\kappa_2$ is the least with Mitchell order 2. Another main theorem is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, with $\kappa_1$ the least measurable cardinal such that $o(\kappa_1) = 2$ and $\kappa_2$ the least measurable cardinal above $\kappa_1$. This is a joint work in progress with Victoria Gitman and Arthur Apter.

Video

This talk is about the ongoing investigation of the relationships between measurable and strongly compact cardinals. I will present some of the history of the theorems in this theme, including Magidor's identity crisis, and give new results. The theorems presented are in particular about the relationships between strongly compact cardinals and measurable cardinals of different Mitchell orders. One of the main theorems is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, and where $\kappa_1$ is least with Mitchell order 1, and $\kappa_2$ is the least with Mitchell order 2. Another main theorem is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, with $\kappa_1$ the least measurable cardinal such that $o(\kappa_1) = 2$ and $\kappa_2$ the least measurable cardinal above $\kappa_1$. This is a joint work in progress with Victoria Gitman and Arthur Apter.

Video

Vopenka's Principle (VP) says that for every proper class of structures with the same signature, there is an elementary embedding from one structure in the class to another. An equivalent form of VP says that for every proper class of graphs, there is an embedding from one graph in the class to another; let us denote this form by VP(graphs, embeddings) with the obvious meaning. We can obtain weaker instances of VP by restricting to particular kinds of graphs such as trees, which are connected acyclic graphs, and rayless trees, which are trees with no infinite path. We will show that VP(trees, embeddings) and VP(rayless trees, embeddings) occupy two different places in the large cardinal hierarchy below VP, and that each is equivalent to the existence of certain virtual large cardinals. Namely, we will show that VP(trees, embeddings) is equivalent to the existence of a weakly virtually A-extendible cardinal (as defined by Gitman and Hamkins) for every class A, and VP(rayless trees, embeddings) is equivalent to the existence of what we will call a weakly virtually A-strong cardinal for every class A. For a better-known point of comparison: the former large cardinal hypothesis is stronger than the existence of a remarkable cardinal, whereas the latter is weaker. We will also relate these two instances of VP to other variants of VP such as generic Vopenka's Principle (as defined by Bagaria, Gitman, and Schindler) and generic Weak Vopenka's Principle.

Video

We are going to talk about some inequalities between cardinal characteristics of the continuum. In particular we are going to relate cardinal characteristics pertaining to the convergenve of series, recently isolated by Blass, Brendle, Brian and Hamkins, other characteristcs concerning equitable splitting defined comparatatively recently by Brendle, Halbeisen, Klausner, Lischka and Shelah and some characteristics defined less recently by Miller, Blass, Laflamme and Minami. All proofs in question are elementary.

Slides

There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object. In contrast, failure of SCH is an instance of incompactness.

Two classical results of Magidor are:
(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and
(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.

As these principles are at odds with each other, the natural question is whether we can have both. We show the answer is yes.

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at $\aleph_\omega$ by interleaving collapses. This is joint work with Alejandro Poveda and Assaf Rinot.

Video

The Triangular Embedding Theorem gives a sort of internal generic absoluteness principle that holds under determinacy or large-cardinal assumptions. It originated in work (joint with Itay Neeman) on mad families and the Ramsey Property under AD^+. I will discuss these origins, some applications, and some questions.

Video

A set x of ordinals is called recognisable if it is defined, as a singleton, by a formula phi(y) with ordinal parameters that is evaluated in L[y]. The evaluation is always forcing absolute, in contrast to even Sigma_1-formulas with ordinal parameters evaluated in V. Furthermore, this notion is closely related to similar concepts in infinite computation and Hamkins' and Leahy's implicitly definable sets.

It is conjectured that the recognisable universe generated by all recognisable sets is forcing absolute, given sufficient large cardinals. Our goal is thus to determine the recognisable universe in the presence of large cardinals. The new main result, joint with Philip Welch, is a computation of the recognisable universe within the least inner model with infinitely many measurable cardinals.

Video

The classical concept of independence, first introduced by Fichtenholz and Kantorovic has been of interest within the study of combinatorics of the subsets of the real line. In particular the study of the cardinal characteristic $\mathfrak{i}$ defined as the minimum size of a maximal independent family of subsets of $\omega.$ In the first part of the talk, we will review the basic theory, as well as the most important results regarding the independence number. We will also point out our construction of a poset $\mathbb{P}$ forcing a maximal independent family of minimal size which turns out to be indestructible after forcing with a countable support iteration of Sacks forcing.

In the second part, we will talk about the generalization (or possible generalizations) of the concept of independence in the generalized Baire spaces, i.e. within the space $\kappa^\kappa$ when $\kappa$ is a regular uncountable cardinal and the new challenges this generalization entails. Moreover, for a specific version of generalized independence, we can have an analogous result to the one mentioned in the paragraph above.

This is joint work with Vera Fischer.

Video

Theorem (Mitchell and Schimmerling, submitted for publication) Assume there is no transitive class model of ZFC with a Woodin cardinal. Let $\nu$ be a singular ordinal such that $\nu > \omega_2$ and $\mathrm{cf}(\nu) < | \nu |$. Suppose $\nu$ is a regular cardinal in K. Then $\nu$ is a measurable cardinal in K. Moreover, if $\mathrm{cf}(\nu) > \omega$, then $o^\mathrm{K}(\nu) \ge \mathrm{cf}(\nu)$.

I will say something intuitive and wildly incomplete but not misleading about the meaning of the theorem, how it is proved, and the history of results behind it.

Video

The real numbers are up to isomorphism the only completely ordered field with a countable dense subset. We consider non-Archimedean ordered fields whose smallest dense subset has cardinality kappa and investigate whether anything resembling ordinary analysis works on these fields.

In particular, we look at generalisations of the intermediate value theorem and the Bolzano-Weierstrass theorem, and realise that there is some mathematical tension between these theorems: the intermediate value theorem requires some saturation whereas Bolzano-Weierstrass fails if the field is saturated. We consider weakenings of Bolzano-Weierstrass compatible with saturation and realise that these are equivalent to the weak compactness of kappa.

This is joint work with Merlin Carl, Lorenzo Galeotti, and Aymane Hanafi.

Video

Countably complete ultrafilters are the combinatorial manifestation of strong large cardinal axioms, but many of their basic properties are undecidable no matter the large cardinal axioms one is willing to adopt. The Ultrapower Axiom (UA) is a set theoretic principle that permits the development of a much clearer picture of countably complete ultrafilters and, consequently, the large cardinals from which they derive. It is not known whether UA is (relatively) consistent with very large cardinals, but assuming there is a canonical inner model with a supercompact cardinal, the answer should be yes: this inner model should satisfy UA and yet inherit all large cardinals present in the universe of sets. The predicted resemblance between the large cardinal structure of this model and that of the universe itself is so extreme as to suggest that certain consequences of UA must in fact be provable outright from large cardinal axioms. While the inner model theory of supercompact cardinals remains a major open problem, this talk will describe a technique that already permits a number of consequences of UA to be replicated from large cardinals alone. Still, the technique rests on the existence of inner models that absorb large cardinals, but instead of building canonical inner models, one takes ultrapowers.

Video

Many large cardinals can be defined through elementary embeddings from the set-theoretic universe to some inner model, with the guiding principle being that the closer the inner model is to the universe the stronger the resulting theory. Under ZFC, the Kunen Inconsistency places a hard limit on how close this can be. One is then naturally led to the question of what theory is necessary to derive this inconsistency with the primary focus having historically been embeddings in ZF without Choice.

In this talk we take a different approach to weakening the required theory, which is to study elementary embeddings from the universe into itself in ZFC without Power Set. We shall see that I1, one of the largest large cardinal axioms not known to be inconsistent with ZFC, gives an upper bound to the naive version of this question. However, under reasonable assumptions, we can reobtain this inconsistency in our weaker theory.

Video

Guzmán and Kalajdzievski introduced a variant of Miller forcing $P(F)$ that diagonalises a given filter $F$ over $\omega$ and has Axiom A. We investigate the effect of $P(F)$ for particularly chosen Canjar filters $F$. This is joint work with Christian Bräuninger.

Video

This will be a sequel to Ralf Schindler’s talk on 9/25. My plan is to give a reasonably detailed account of the proof of the result in the title.

Video

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and 'consistent' needs to mean 'consistent in a strong sense.' It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. This is joint work with D. Asperó building upon earlier work of R. Jensen and (ultimately) Keisler's 'consistency properties'.

Video

The Ultrapower Axiom UA, introduced by Goldberg and Woodin, is known to have many striking consequences. In particular, Goldberg has shown that assuming UA, the Mitchell ordering of normal measures over a measurable cardinal is linear. I will discuss how this result may be used to construct choiceless models of ZF in which the number of normal measures at successors of singular cardinals can be precisely controlled.

Slides

Video

We study logics which model the passage between an infinite sequence of finite models to an uncountable limiting object, such as is the case in the context of graphons. Of particular interest is the connection between the countable and the uncountable object that one obtains as the union versus the combinatorial limit of the same sequence.

Video

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.