In 2001, Mancini and Zambella investigated computable models of fragments of set theory. They emplyed the Bernays-Rieger method of permutations to construct a computable model of finite set theory (i.e. $ZF_{fin}$ – the theory obtained from ZF by replacing the axiom of infinity by its negation). In 2009, Enayat, Schmerl and Visser showed how to build computable nonstandard models of this theory without the use of permutations. Furthermore, they demonstrated that in every computable model of ZF_{fin} every set (as viewed externally) has only finitely many elements (such models are called $\omega$-models). The corollaries of these results are, among others, that there are continuum-many nonisomorphic pointwise definable $\omega$-models of $ZF_{fin}$ and that $PA$ and $ZF_{fin}$ are not bi-interpretable. The purpose of the talk is to present proofs of the results of Macinini, Zambella and Enayat, Schmerl, Visser together with the corollaries.

In 2010, a question on MathOverflow asked whether it is possible that a countable ordinal definable set of reals has elements that are not ordinal definable. It is easy to see that every element of a finite ordinal definable set of reals is itself ordinal definable. Also it is consistent that there is an uncountable ordinal definable set of reals without ordinal definable elements. It turned out that the question for countable sets of reals was not known. It was finally solved by Kanovei and Lyubetsky in 2014, who showed, using a forcing extension by a finite-support product of Jensen reals, that it is consistent to have a countable ordinal definable set of reals without ordinal definable elements. In the talk, I will give full details of their argument.

An extended abstract is available on my blog here.

For an infinite cardinal $\lambda$, we say that $\lambda$ is Jónsson if it satisfies the square bracket partition property $\lambda\to[\lambda]^{<\omega}_\lambda$. This means that, for every coloring $F\colon [\lambda]^{<\omega}\to\lambda$, there is some set $H\in [\lambda]^\lambda$ with the property that $\mathrm{ran}(F\upharpoonright [H]^{<\omega})\subsetneq\lambda$. It is rather well known that the consistency strength of "there is a Jónsson cardinal" lies above "$0^\sharp$ exists", but below the existence of a measurable. However, the question of what sorts of cardinals can be Jónsson has turned out to be rather difficult. The goal of this talk is to sketch a simplified proof of the result (due to Shelah) that if $\lambda$ is the least regular Jónsson cardinal, then $\lambda$ must be $\lambda\times\omega$-Mahlo. The advantage of the proof presented here is that the machinery employed can be easily generalized, and time permitting I would like to discuss how one might attempt to prove that such a cardinals must be greatly Mahlo.

I shall present a proof of a theorem of Bukovský from 1973 that characterizes the set-forcing extensions among all pairs of ZFC models $M\subseteq N$: these are precisely the pairs satisfying a uniform covering property. His result has recently resurfaced in the study of set-theoretic geology and can, for example, also be used to give a conceptual proof of (a version of) the intermediate model theorem.

The method of matrix iterations was introduced by Blass and Shelah in their study of the dominating and the ultrafilter numbers. Since its appearance, the method has undergone significant development and applied to the study of many other cardinal characteristics of the continuum, including those associated to measure and category.

Recently, we were able to extend the technique of matrix iterations to a “third dimension” and so, evaluate the almost disjointness number in models where previously its value was not known. In addition, we obtain new constellations of the Cichon diagram (with up to seven distinct values). This is a joint work with Friedman, Mejia and Montoya.

In 1972 Vopenka showed that $V$ is a union of set-generic extensions of $\mathrm{HOD}$ by establishing that every set in $V\setminus\mathrm{HOD}$ is set generic over $\mathrm{HOD}$. It is natural to consider whether that union can be replaced by a single forcing, possibly a proper class, over $\mathrm{HOD}$. In 2012 Friedman showed that $V$ is a class forcing extension of $\mathrm{HOD}$ by a partial order definable in $V$ – however, this leaves open the question of whether such a partial order can be defined in $\mathrm{HOD}$ itself. In this talk I will show that the qualifier ‘in $V$’ is necessary in Friedman’s theorem, by producing a model which is not class generic over $\mathrm{HOD}$ for any forcing definable in $\mathrm{HOD}$.

In the area of set theory known as set-theoretic geology, the mantle $M$ (the intersection of all grounds) is an inner model that enjoys a relationship to $V$ similar to that of $\mathrm{HOD}$, but ‘in the opposite direction’ – every set not in $M$ is omitted by a ground of $V$. Does it follow that we can build $V$ up over $M$ by iteratively adding those sets back in via forcing? In particular, does it follow that $V$ is a class forcing extension of $M$? The example produced in this talk will show that the answer is no – there is a model of set theory $V$ which is not a class forcing extension of $M$ by any forcing definable in $M$.

I’ll begin by describing two fascinating questions in ergodic theory, one being the isomorphism problem for measure-preserving transformations. I’ll survey some of the progress that has been made on these problems, including some partial solutions to the isomorphism problem for certain classes of measure-preserving transformations and some anti-classification results, stating that “nice” solutions to the isomorphism problem are impossible on other classes of measure-preserving transformations. Then I’ll discuss recent work I’ve done with Su Gao on the isomorphism problem for the class of rank-1 transformations, a generic class of measure-preserving transformations where the isomorphism relation is known to be, in some sense, well-behaved. (Background information ergodic theory will be introduced as needed.)

In his dissertation, Steel showed that both open determinacy and clopen determinacy have a reverse math strength of $\mathsf{ATR}_0$. In particular, this implies that clopen determinacy is equivalent to open determinacy (over a weak base theory). Gitman and Hamkins in recent work explored determinacy for class-sized games in the context of second-order set theory. They asked whether the analogue of Steel’s result holds: over $\mathsf{GBC}$, is open determinacy for class games equivalent to clopen determinacy for class games?

The answer, perhaps surprisingly, is no. In this talk I will present some recent results by Hachtman which answer the question of Gitman and Hamkins. We will see how to construct a transitive model of $\mathsf{GBC}$ which satisfies clopen class determinacy but does not satisfy open class determinacy.

${\rm AD}_{\mathbb R}$ is a strengthening of the determinacy axiom that states that all games on the real numbers are determined. It is a Theorem of Solovay that under ${\rm ZF}+{\rm AD}_{\mathbb R}$ there is a fine, countably complete and normal filter on $P_{\omega_1}(\mathbb R)$, so $\omega_1$ is $\mathbb R$-supercompact. The exact consistency strength of the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact” is, however, weaker than the one of ${\rm ZF}+{\rm AD}_{\mathbb R}$.
One central interest of Inner Model Theory is to construct/find canonical models for theories extending ${\rm ZF}$. A natural question is, then, whether there is a canonical model for the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact”.
In this talk, we will discuss the consistency strength and minimal models of this theory. We will discuss the proof of the uniqueness of minimal models of this theory, under various appropriate hypotheses. And time permitting we will discuss the proof of the result that under ${\rm ZFC}$ there is at most one minimal model of this theory. This is joint work with Nam Trang.

We show that, in many cases, there is a Borel reduction from the isomorphism relation on a given Fraïssé class to the conjugacy relation on the automorphism group of the Fraïssé limit. Hence, if the former is Borel complete, then so is the latter. The key property is a functorial, Borel form of amalgamation. All relevant notions about Borel redicibility and Fraïssé classes will be defined.

By an old result of Mathias, there are no mad families in the Solovay model constructed by the Levy collapse of a Mahlo cardinal. By a recent result of Törnquist, the same is true in the classical model of Solovay as well. In a recent paper, we show that ZF+DC+”there are no mad families” is actually equiconsistent with ZFC. I’ll present the ideas behind the proof in the first part of the talk.

In the second part of the talk, I’ll discuss the definability of maximal eventually different families and maximal cofinitary groups. In sharp contrast with mad families, it turns out that Borel MED families and MCGs can be constructed in ZF. Finally, I’ll present a general problem in Borel combinatorics whose solution should explain the above difference between mad and maximal eventually different families, and I’ll show how large cardinals must be involved in such a solution.

This is joint work with Saharon Shelah.

I will continue presenting Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. See the main abstract at Set-theoretic geology and the downward directed ground hypothesis.

See my blog post about this talk.

Forcing is often viewed as a method of constructing larger models extending a given model of set theory. The topic of set-theoretic geology inverts this perspective by investigating how the current set-theoretic universe $V$ might itself have arisen as a forcing extension of an inner model. Thus, an inner model $W\subset V$ is a ground of $V$ if we can realize $V=W[G]$ as a forcing extension of $W$ by some $W$-generic filter $G\subset\mathbb Q\in W$. Reitz had inquired in his dissertation whether any two grounds of $V$ must have a common deeper ground. Fuchs, myself and Reitz introduced the downward-directed grounds hypothesis, which asserts a positive answer, even for any set-indexed collection of grounds, and we showed that this axiom has many interesting consequences for set-theoretic geology.

I shall give a complete detailed account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. This breakthrough result answers what had been for ten years the central open question in the area of set-theoretic geology and leads immediately to numerous consequences that settle many other open questions in the area, as well as to a sharpening of some of the central concepts of set-theoretic geology, such as the fact that the mantle coincides with the generic mantle and is a model of ZFC. I shall also present Usuba’s related result that if there is a hyper-huge cardinal, then there is a bedrock model, a smallest ground. I find this to be a surprising and incredible result, as it shows that large cardinal existence axioms have consequences on the structure of grounds for the universe.

See my blog post about this talk.

One of the most fruitful research area in set theory is the study of the so-called Reflections Principles. Roughly speaking, a reflection principle is a combinatorial statement of the following form: given a structure S (e.g. stationary sets, tree, graphs, groups …) and a property P of the structure, the principle establishes that there exists a smaller substructure of S that satisfies the same property P. There is a tension between large cardinals axioms and the axiom of constructibility V=L at the level of reflection: on the one hand, large cardinals typically imply reflection properties, on the other hand L satisfies the square principles which are anti-reflection properties. Two particular cases of reflection received special attention, the reflection of stationary sets and the tree property. We will discuss the interactions between these principles and a version of the square due to Todorcevic. This is a joint work with Menachem Magidor and Yair Hayut.

This talk is a more detailed version of the presentation that I gave on set theory day.

I present a position in infinite chess with game value $\omega^4$. Informally speaking, this means that one side can force a win in $\omega^4$ many moves, or that the game is equivalent to the game of counting down from $\omega^4$. This result, joint with Hamkins and Evans, improved on the previous best result of $\omega^3$.

I will discuss the number of normal measures a non-$(\kappa + 2)$-strong tall cardinal $\kappa$ can carry, paying particular attention to the cases where $\kappa$ is either the least measurable cardinal or the least measurable limit of strong cardinals. This is joint work with James Cummings.

Slides

We’ll discuss problems arising when trying to apply CMI in models where even the weakest forms of choice might fail. We’ll show how to deal with these problems in the particular case of a model in which all uncountable cardinals are singular.