Set Theory Talks

Speaker:     Evan Misshula, CUNY.

Date and Time:     Wednesday December 3, 2025, 7:00 - 8:30 PM. IN-PERSON TALK, Room 6417

Title:     From Outer Measures to Adjunctions: A Category-Theoretic Recasting of Caratheodory’s Extension Theorem.

Abstract: The Caratheodory Extension Theorem underpins modern measure theory by extending a pre-measure on an algebra to a complete measure on the generated -algebra. Traditionally, the proof proceeds through outer measures, Caratheodory measurability, and a series of delicate technical verifications – a process that reveals the "monsters" lurking in the power set but often obscures the structural simplicity of the result. In this talk, I will first outline the classical construction to highlight these subtleties, and then present a category-theoretic reformulation: the extension theorem emerges naturally from an adjunction. The categorical perspective streamlines the argument, but seeing both sides illuminates why rigor is indispensable and how category theory captures the essence of the construction.

Mostafa Mirabi, Wesleyan University
Asymptotic Classes of Trees and ℵ0-Categoricity

In this talk we introduce a framework for analyzing ℵ0-categorical theories of trees using a combinatorial notion called a tree plan. We show that every ℵ0-categorical theory of trees arises from a Fraïssé class K determined by a tree plan, and that each such K forms an asymptotic class in the sense of Macpherson--Steinhorn. We then discuss the model-theoretic consequences of this perspective, including the behavior of ultraproducts, the emergence of ℵ0-categoricity from asymptotic classes of finite trees, and a characterization of supersimple finite-rank theories of trees in terms of their associated tree plans.

Speaker:     Evan Misshula, CUNY.

Date and Time:     Wednesday December 3, 2025, 7:00 - 8:30 PM. IN-PERSON TALK, Room 6417

Title:     From Outer Measures to Adjunctions: A Category-Theoretic Recasting of Caratheodory’s Extension Theorem.

Abstract: The Caratheodory Extension Theorem underpins modern measure theory by extending a pre-measure on an algebra to a complete measure on the generated -algebra. Traditionally, the proof proceeds through outer measures, Caratheodory measurability, and a series of delicate technical verifications – a process that reveals the "monsters" lurking in the power set but often obscures the structural simplicity of the result. In this talk, I will first outline the classical construction to highlight these subtleties, and then present a category-theoretic reformulation: the extension theorem emerges naturally from an adjunction. The categorical perspective streamlines the argument, but seeing both sides illuminates why rigor is indispensable and how category theory captures the essence of the construction.

Mostafa Mirabi, Wesleyan University
Asymptotic Classes of Trees and ℵ0-Categoricity

In this talk we introduce a framework for analyzing ℵ0-categorical theories of trees using a combinatorial notion called a tree plan. We show that every ℵ0-categorical theory of trees arises from a Fraïssé class K determined by a tree plan, and that each such K forms an asymptotic class in the sense of Macpherson--Steinhorn. We then discuss the model-theoretic consequences of this perspective, including the behavior of ultraproducts, the emergence of ℵ0-categoricity from asymptotic classes of finite trees, and a characterization of supersimple finite-rank theories of trees in terms of their associated tree plans.

Title: The 85th Nankai Logic Colloquium--Alberto Marcone Time: 16:00,November.21, 2025(Beijing Time) Zoom Number: 258 372 3976 Passcode: 304007 Link: https://zoom.us/j/2583723976?pwd=avtTJchXaFFVTKK4u6GOh61xhoZus2.1&omn=99618102698   

Aris Papadopoulos, University of Maryland
Zarankiewicz's Problem and Model Theory

A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: 'How 'sparse' must a given bipartite graph be, if I know that it has no 'dense' subgraphs?'. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk.

So far so good, but this is a logic workshop and the title says the words 'Model Theory'… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz's problem in 'geometric' contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz's problem, when we restrict our attention to semilinear/semibounded o-minimal structures, Presburger arithmetic, and various kinds of Hrushovski constructions. The new results that will appear in the talk have been obtained jointly with Pantelis Eleftheriou.

Dan Turetsky, Victoria University of Wellington
Iterated derivatives and antiderivatives of graphs

The Hausdorff derivative of a linear order can be iterated to an ordinal length, giving a sequence of quotient linear orders, where each step requires a double jump to calculate.. Ash and Watnick give a converse to this, where the antiderivatives are product orderings of an appropriate lower complexity than the original ordering. Motivated by uncountable computability theory, we wanted a variant of this in which the derivatives are substructures rather than quotient structures. Once we had this, it turned out to apply not just to linear orders, but also graphs, trees and forests. I will explain our theorem primarily in the context of countable graphs and computability theory, but with some asides about other structures and uncountable computability theory.

Adding Isomorphisms between Dense Sets of Reals

In this talk we will discuss some problems regarding adding linear order isomorphisms between dense sets of reals via forcing. A set of reals A⊆R is called ℵ1-dense just in case it has intersection size ℵ1 with every nonempty open interval. Famously Baumgartner showed in the 70s that consistently all ℵ1-dense sets of reals are order isomorphic, thus establishing the consistency of the natural analogue of Cantor's categoricity theorem for countable dense linear orders at the uncountable. He later showed the same statement follows from PFA. The statement 'all ℵ1-dense sets of reals are isomorphic' is now known as Baumgartner's axiom and denoted BA.

Baumgartner's argument is famously tricky and has several interesting features. Given ℵ1-dense sets A and B he shows that under the continuum hypothesis there is always a ccc partial order for making them isomorphic - but here the CH is important (though it must fail in the final model). Obviously it is therefore natural to ask whether the CH is necessary and, similarly whether BA follows already from MA and not just PFA. Avraham and Shelah showed the answer is 'no' to both: they produced a model of MA in which there is an ℵ1-dense set of reals A so that no ccc forcing notion can add an isomorphism between A and its reverse ordering. In the first part of this talk we will strengthen this result by showing that MA is in fact consistent with an ℵ1-dense linear order A so that any partial order of size ℵ1 which adds an isomorphism between A and its reverse ordering must collapse ℵ1. Thus it is consistent with MA that no such order can even be proper. This part is joint work with Pedro Marun and Saharon Shelah.

In another direction there is a natural generalization of BA to non-ordered spaces, most notably higher dimensional Euclidean spaces Rn for n>1 as well as compact n-dimensional manifolds. Curiously, in these cases Steprans and Watson showed that the corresponding BA statements do indeed follow from MA so the case of dimension one is unique. In particular BA for Rn with n>1 does not imply the one dimensional case. They then conjectured that conversely BA implies its higher dimensional analogues. In the second part of the talk we will introduce some intrigue to this conjecture by showing any 'reasonable' way of forcing BA - a very general adjective that includes all known methods - must necessarily force the higher dimensional versions and much more including large fragments of MA.

Levels of projective determinacy and levels of dependent choice

Gitman, Friedman, and Kanovei described a construction of a model of ZF in which countable choice holds and Π12-dependent choice for reals is not satisfied. We modify that construction to obtain a model in which (boldface) Π1n-determinacy holds, but which fails to satisfy (lightface) Π1n+2-DC for reals. In particular, we show that no projective level of determinacy implies full DCR. This is joint work with Sandra Müller.

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.

Daniel Isaacson, Oxford University
Consideration of Dummett's claim that the meaning of 'natural number' is inherently vague

Invariant Cuts of Countable Short Recursively Saturated Models of PA

In this talk we continue the discussion on the automorphism groups of countable short recursively saturated models of PA. In particular, we discuss the cuts of the model which are fixed setwise by all automorphisms (invariant cuts). We show that such cuts occur in different places of the model, depending on the types realized in the last gap. We then show that this implies, in some of these cases, that the automorphism groups of such models are non-isomorphic as topological groups. This is a joint work with Ermek Nurkhaidarov.

Daniel Isaacson, Oxford University
Consideration of Dummett's claim that the meaning of 'natural number' is inherently vague

Invariant Cuts of Countable Short Recursively Saturated Models of PA

In this talk we continue the discussion on the automorphism groups of countable short recursively saturated models of PA. In particular, we discuss the cuts of the model which are fixed setwise by all automorphisms (invariant cuts). We show that such cuts occur in different places of the model, depending on the types realized in the last gap. We then show that this implies, in some of these cases, that the automorphism groups of such models are non-isomorphic as topological groups. This is a joint work with Ermek Nurkhaidarov.

Daniel Isaacson, Oxford University
Consideration of Dummett's claim that the meaning of 'natural number' is inherently vague

Stefan Hoffelner TU Wien
The global Σ1n+2-Uniformization Property and BPFA

We show that, given a reflecting cardinal, one can generically produce a universe of BPFA in which additionally the Σ1n+2-uniformization property holds for every n simultaneously.

Examples of Distal Metric Structures

We identify several examples of distal metric structures and examine several consequences of distality, such as the existence of distal cell decompositions, in each. These results include joint work with Itaï Ben Yaacov and with Diego Bejarano. One class of examples starts with finding a metric structure whose automorphism group is the group of increasing homeomorphisms of the unit interval. We will discuss some properties of this structure and extrapolate to other models of its theory, which we call 'dual linear continua.' Another source of examples includes real closed metric valued fields. These give rise to a notion of ordered metric structure, providing a viewpoint to study o-minimality in continuous logic.