Set Theory Seminar

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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'.

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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

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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.

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