Set Theory Seminar

1. I'll begin by reviewing the work I did in 1993-6 on a problem raised by the dynamics group at the Universidad Autonomoa de Barcelona. They were interested in a phenomenon that resembles the Cantor-Bendixson sequence of derivatives, and hoped to prove that the sequence would always stop at a countable stage. Using ideas of Kunen and Martin I showed that it would always stop at or before stage omega_1.

2. In 2002/3, alerted by observations of David Fremlin, to the possibility that the barcelona conjecture was false, I succeeded in constructing an example with recursive initial data where the sequence stops exactly at stage omega_1.

My Réunion colleague Chrstian Delhommé simplified and extended my ideas.

I'll outline the construction, as I think the underlying idea might have applications elsewhere in descriptive set theory.

3. I will outline more recent work using ideas of Blass and Fremlin to to study 'uniform' versions of the results of 1993-96.

I'll end with listing some open problems which I hope will be found interesting.

References

Slides

Video

1. I'll begin by reviewing the work I did in 1993-6 on a problem raised by the dynamics group at the Universidad Autonomoa de Barcelona. They were interested in a phenomenon that resembles the Cantor-Bendixson sequence of derivatives, and hoped to prove that the sequence would always stop at a countable stage. Using ideas of Kunen and Martin I showed that it would always stop at or before stage omega_1.

2. In 2002/3, alerted by observations of David Fremlin, to the possibility that the barcelona conjecture was false, I succeeded in constructing an example with recursive initial data where the sequence stops exactly at stage omega_1.

My Réunion colleague Chrstian Delhommé simplified and extended my ideas.

I'll outline the construction, as I think the underlying idea might have applications elsewhere in descriptive set theory.

3. I will outline more recent work using ideas of Blass and Fremlin to to study 'uniform' versions of the results of 1993-96.

I'll end with listing some open problems which I hope will be found interesting.

References

Slides

Video

It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that the continuum is a proper class. We examine several theories based on maximality considerations in this framework (in particular countabilist analogues of reflection principles) and show how standard set theories (including ZFC with large cardinals added) can be incorporated. We conclude that the systems considered raise questions concerning the foundational purposes of set theory.

Video

We continue some work on L[Card] (the universe constructed from the predicate for the cardinals) to look at L[Reg] where Reg is the class of uncountable regular cardinals. The latter is also a model of a rich combinatorial structure being, as it turns out, a Magidor iteration of prikry forcings (using recent work of Ben-Neria). But it is limited in size, in fact is a rather 'thin' model. We show, letting O^s = O^sword be the least iterable structure with a measure which concentrates on measurable cardinals:

Theorem (ZFC)

1. Let S be a set, or proper class, of regular cardinals, then O^s is not an element of L[S].
2. (b) This is best possible, in that no smaller mouse M can be substituted for O^s.
3. (c) L[S] is a model of: GCH, Square's, Diamonds, Morasses etc and has Ramsey cardinals, but no measurable cardinals.

Slides

Video

In a recent Quanta Magazine article discussing difficulties and progress related to Feynman path integrals, Charlie Wood writes, 'No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general.' This statement is arguably refuted by Nonstandard Analysis, but what is perhaps lacking is a constructive approach. We present such an approach based on reduced powers and a class of algebraic structures we call comparison rings. This construction has a nice iteration theory and is able to represent classical integrals via standard parts. We discuss an example of a filter on $\mathbb R^{\lt\omega}$, the direct limit of the $\mathbb R^n$, that respects classical volumes in different dimensions simultaneously, with lower dimensional surfaces being infinitesimal relative to higher dimensional ones. This suggests a corresponding generalization of dimension, which we show under some set-theoretic assumptions may constitute a dense linear order without $(c,c)$-gaps. This is joint work with Emanuele Bottazzi.

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

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

Woodin's HOD conjecture asserts that in the context of very large cardinals, the inner model HOD closely approximates the universe of sets in the same way Gödel's constructible universe does assuming 0# does not exist. The subject of these two talks is the relationship between Woodin's conjecture and certain constraints on the structure of elementary embeddings of the universe of sets. For example, in the second talk, we will prove that any two elementary embeddings of the universe of sets into the same inner model agree on HOD, while if a local version of this theorem held, then the HOD conjecture would follow.

Video

Woodin's HOD conjecture asserts that in the context of very large cardinals, the inner model HOD closely approximates the universe of sets in the same way Gödel's constructible universe does assuming 0# does not exist. The subject of these two talks is the relationship between Woodin's conjecture and certain constraints on the structure of elementary embeddings of the universe of sets. For example, in the second talk, we will prove that any two elementary embeddings of the universe of sets into the same inner model agree on HOD, while if a local version of this theorem held, then the HOD conjecture would follow.

Video

In his seminal work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals known as the Magidor iteration. The purpose of this talk is to present a Mathias-type criterion which characterizes when a sequence of omega-sequences is generic for the Magidor iteration. The result extends a theorem of Fuchs, who introduced a Mathias criterion for discrete products of Prikry forcings. We will present the new criterion, discuss several applications, and outline the main ideas of the proof.

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.

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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

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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