Set Theory Seminar

We survey some old and new results in singular cardinal combinatorics whose proofs can be phrased in terms of iterated ultrapowers and ask a few questions.

Lindstrom Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the downward Lowenheim-Skolem Theorem. Despite intensive efforts, no such model theoretic characterizations were obtained for infinitary logics, until Shelah introduced his logic $L^1_\kappa$. We define a new class of logics $L^1_{\kappa,\alpha}$ generalizing Shelah's logic and examine their expressive power. If $\kappa$ is a fixed point of the $\beth$-function these logics coincide with $L^1_\kappa$. We given a different version of Lindstrom's theorem in terms of the $\phi$-submodel relation.

We also discuss algebraic characterization of the elementary equivalence relation for these logics for suitable $\kappa$.

Joint work with Jouko Väänänen.

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In the late 1910s Bertrand Russell was occupied with two things: getting into political trouble for his pacifism and trying to understand the foundations of mathematics. His students were hard at work with him on this second occupation. One of those students was Dorothy Wrinch. In 1923 she gave a characterization of the axiom of choice in terms of a generalization of the notion of a Dedekind-finite infinite set. Unfortunately, her career turned toward mathematical biology and her logical work was forgotten by history.

This talk is part of a project of revisiting Wrinch's work from a modern perspective. I will present the main result of her 1923 paper, that AC is equivalent to the non-existence of what she termed mediate cardinals. I will also talk about some new independence results. The two main results are: (1) the smallest $\kappa$ for which a $\kappa$-mediate cardinal exists can consistently be any regular $\kappa$ and (2) the collection of regular $\kappa$ for which exact $\kappa$-mediate cardinals exist can consistently be any class.

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In a lecture presented in July 2023, Moti Gitik discussed the following question from the 1980s due to Woodin, as well as approaches to its solution and why it is so difficult to solve:

Question: Assuming there is no inner model of ZFC with a strong cardinal, is it possible to have a model $M$ of ZFC such that $M \vDash$'$2^{\aleph_\omega} > \aleph_{\omega + 2}$ and $2^{\aleph_n} = \aleph_{n + 1}$ for every $n \lt\omega$', together with the existence of an inner model $N^* \subseteq M$ of ZFC such that for the $\gamma, \delta$ so that $\gamma = (\aleph_\omega)^M$ and $\delta = (\aleph_{\omega + 3})^M,$ $N^* \vDash$'$\gamma$ is measurable and $2^\gamma \ge \delta$'?

I will discuss how to find answers to this question, if we drop the requirement that $M$ satisfies the Axiom of Choice. I will also briefly discuss the phenomenon that on occasion, when the Axiom of Choice is removed from consideration, a technically challenging question or problem becomes more tractable. One may, however, end up with models satisfying conclusions that are impossible in ZFC.

Reference: A. Apter, 'A Note on a Question of Woodin', Bulletin of the Polish Academy of Sciences (Mathematics), volume 71(2), 2023, 115--121.

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A uniform ultrafilter $U$ over a cardinal $\kappa > \omega_1$ is called indecomposable if, whenever $\lambda \lt\kappa$ and $f:\kappa \rightarrow \lambda$, there is a set $X \in U$ such that $f[X]$ is countable. Indecomposability is a natural weakening of $\kappa$-completeness and has a number of implications for, e.g., the structure of ultraproducts. In the 1980s, Sheard answered a question of Silver by proving the consistency of the existence of an inaccessible but not weakly compact cardinal carrying an indecomposable ultrafilter. Recently, however, Goldberg proved that this situation cannot hold above a strongly compact cardinal: If $\lambda$ is strongly compact and $\kappa \geq \lambda$ carries an indecomposable ultrafilter, then $\kappa$ is either measurable or a singular limit of countably many measurable cardinals. We prove that the same conclusion follows from the Proper Forcing Axiom, thus adding to the long list of statements first shown to hold above a strongly compact or supercompact cardinal and later shown also to follow from PFA. Time permitting, we will employ certain indexed square principles to prove that our results are sharp. This is joint work with Assaf Rinot and Jing Zhang.

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We consider logics in which the collection of sentences over a set-sized vocabulary can form a proper class. The easiest example of such a logic is $\mathcal L_{\infty \infty}$, which allows for disjunctions and conjunctions over arbitrarily sized sets of formulas and quantification over strings of variables of any infinite length. Model theory of $\mathcal L_{\infty \infty}$ is very restricted. For instance, it is inconsistent for it to have nice compactness or Löwenheim-Skolem properties. However, Trevor Wilson recently showed that the existence of a Löwenheim-Skolem-Tarski number of a certain class-sized fragment of $\mathcal L_{\infty \infty}$ is equivalent to the existence of a supercompact cardinal, and various other related results. We continue this work by considering several appropriate class-sized logics and their relations to large cardinals. This is joint work with Trevor Wilson.

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In the first part of the talk, we will provide some background and motivation to study the Glavin property. In particular, we will present a recently discovered connection between the Galvin property and the Tukey order on ultrafilters. This is a joint result with Natasha Dobrinen. In the second part, we will introduce several diamond-like principles for ultrafilters, and prove some relations with the Galvin property. Finally, we use the Ultrapower Axiom to characterize the Galvin property in the known canonical inner models. The second and third part is joint work with Gabriel Goldberg.

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Mutual stationarity is a compactness type property for singular cardinals. Roughly, it asserts that given a singular cardinal $\kappa$, stationary subsets of regular cardinals with limit $\kappa$ have a 'simultaneous witness' for their stationarity. This principle was first defined by Foreman and Magidor in 2001, who showed that it holds when the stationary sets are of points of countable cofinality. They also showed that in general this does not generalize to higher cofinality. Whether the principle can consistently hold for higher cofinalities remained open, until a few years ago Ben Neria showed that from large cardinals mutual stationarity at $\langle\aleph_n\mid n\lt\omega\rangle$ can be forced for any fixed cofinality.

We show that we can obtain mutual stationarity at $\langle\aleph_n\mid n\lt\omega\rangle$ for any fixed cofinality together with the failure of SCH at $\aleph_\omega$. Along the way we reduce the Ben Neria's large cardinal hypothesis. This is joint work with Will Adkisson.

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