Set Theory Talks

Cecelia Higgins, Rutgers University
Complexity of finite Borel asymptotic dimension

A Borel graph is hyperfinite if it can be written as a countable increasing union of Borel graphs with finite components. It is a major open problem in descriptive set theory to determine the complexity of the set of hyperfinite Borel graphs. In a recent paper, Conley, Jackson, Marks, Seward, and Tucker-Drob introduce the notion of Borel asymptotic dimension, a definable version of Gromov's classical notion of asymptotic dimension, which strengthens hyperfiniteness and implies several nice Borel combinatorial properties. We show that the set of locally finite Borel graphs having finite Borel asymptotic dimension is Σ12-complete. This is joint work with Jan Grebik.

Hans Schoutens, CUNY
Can categories categorize the theories of model-theory?

I want to argue that when knowing the model-theory of categories, you kind of know the model-theory of any structure. As the ? at the end of the title suggests, some of this is still speculative.

It is easy to see a category as a first-order structure in the two-sorted language (for objects and morphisms) of categories; a little less to do this foundationally correct (I have given a talk a way back in which I ignored these issues, but I will correct this in the talk, although not mentioning them in this abstract). Now, to any theory T in some first-order language L, we can associate a theory in the language of categories, cat(T), which reflects this theory: the models of cat(T) are isomorphic (as categories) with subcategories of the category Mod(T) of models of T. In fact, any category that is elementary equivalent with Mod(T) is a sub-model of the latter.

This translation from T into cat(T)---from an arbitrary signature to a fixed one---is still mysterious, and as of now, I only know a very few concrete cases. A key role seems to be played by the theory FO, consisting of all sentences in the language of categories which hold in each category of L-structures, for all possible languages L. But I do not even know yet a full axiomatization of FO.

• Fernando Cano-Jorge (University of Otago)
• Thomas Macaulay Ferguson (Rensselaer Polytechnic Institute)
• Graham Priest (CUNY Graduate Center)

Eyal Kaplan, University of California, Berkeley
The number of normal measures, revisited

A central question in the theory of large cardinals was whether the existence of a model of ZFC with exactly two normal measures follows from the consistency of ZFC with a measurable cardinal. This was answered positively by a landmark theorem of Friedman and Magidor, whose proof masterfully combined advanced techniques in the theory of large cardinals, including generalized Sacks forcing, forcing over canonical inner models, coding posets, and nonstationary support iterations.

In this talk, we present a new and simpler proof of the Friedman-Magidor theorem. A notable feature of our approach is that it avoids any use of inner model theory, making it applicable in the presence of very large cardinals that are beyond the current reach of the inner model program. If time permits, we will also discuss additional applications of the technique: the construction of ZFC models with several normal measures but a single normal ultrapower; a nontrivial model of the weak Ultrapower Axiom from the optimal large cardinal assumption; and a generalization of the Friedman–Magidor theorem to extenders.

Rahman Mohammadpour, Institute of Mathematics of Polish Academy of Sciences
Specializing Triples

I will talk about weak embeddability and the universality number of the class of Aronszajn trees, with a focus on the role of specializing triples.

The notion of a specializing triple was introduced by Džamonja and Shelah in their strong negative solution to an old problem on the existence of a universal (with respect to weak embeddability) wide Aronszajn tree under Martin's axiom. Their proof has two stages: first, they reprove a theorem of Todorčević showing that under MAω1 there is no universal Aronszajn tree, and then they show that every wide Aronszajn tree weakly embeds into an Aronszajn tree. The second stage involves a rather complicated ccc forcing. However, already in the first stage, they introduce a new technique: the notion of a specializing triple, and prove that for each Aronszajn tree T, there is a ccc forcing adding another Aronszajn tree T∗ together with a specializing function on T∗⊗T such that (T∗,T,c) is a specializing triple. In particular, this shows that T∗ does not weakly embed into T.

I will explain how a slight but careful modification of this definition makes it possible to accommodate wide trees directly, yielding a more streamlined proof of Džamonja and Shelah’s result. More precisely, for every κ-wide Aronszajn tree T, there is a ccc forcing adding an Aronszajn tree T∗ and a function c such that (T∗,T,c) is what I call a left specializing triple. From this, one quickly recovers Džamonja-Shelah’s theorem: under Martin’s axiom, every class of trees of height ω1 and size less than the continuum but with no cofinal branches either is not universal for Aronszajn trees, or has universality number equal to the continuum.

Finally, I will indicate how the modified definition can also be used to show that this consequence of Martin’s axiom is consistent with the existence of a nonspecial Aronszajn tree.

Catalina Torres Pachon, University of Barcelona
A Topological Approach to Characterising Hyperstationary Sets on Pκ(A)

Given a topological space (X,τ), the derived set operator dτ maps a set T to its set of limit points with respect to τ. Fixing an initial topology τ0 on X, we can define a sequence of derived topologies (X,τ0,τ1,…,τξ,…), where τζ⊆τξ for ζ<ξ. This is achieved by declaring dτξ(T) to be open in τξ+1 and taking unions at limit stages.

In Derived Topologies on Ordinals and Stationary Reflection, Bagaria characterised the non-isolated points in the ξ-th derived topology on ordinals as those satisfying a strong iterated form of stationary reflection, termed ξ-simultaneous reflection.

Generalisations of combinatorial properties of ordinals to Pκ(A):={X⊆κ:|X|<κ}, where κ is an uncountable regular cardinal and A⊆κ, have been widely studied. In this context, we extend the notion of higher stationarity and construct a sequence of topologies ⟨τ0,τ1,…⟩ on Pκ(A), characterising the simultaneous reflection of high-stationary subsets of Pκ(A) in terms of elements in the base of a derived topology on Pκ(A).

Vince Guingona Towson University

Vince Guingona Towson University

Vince Guingona Towson University