Logic

Woodin's HOD Dichotomy states that in the presence of large cardinals, the universe of sets is either very close to being HOD, or it is very very far from that. In this recent work with Hope Duncan and Gabe Goldberg we show that in the presence of a supercompact cardinal, a similar dichotomy theorem can be proved: either the axiom of choice fails very very badly, or else in many "reasonable inner models", the axiom of choice can be added generically

The logician is familiar with the idea that filters are generalised points. Topos theorists push this idea further by asking: what is a generalised space?

This talk will focus on one such answer: LT-Topologies ("Lawvere-Tierney Topologies"). In the 1980s, Hyland introduced the Effective Topos, and showed that the Turing Degrees embed into its poset of LT-Topologies, what we call the Lawvere-Tierney order (LT-order). Progress was slow until 2013, where Lee-van Oosten showed that this topological framework reveals a meaningful interaction between combinatorics and computable complexity.

In joint work with T. Kihara, we advance this perspective considerably by introducing a game-theoretic version of Katetov's Order. Essentially, this is the usual Katetov order now closed under well-founded iterations of Fubini powers. The first major theorem of our recent paper (arXiv:2602.08138) was to show that a computable variant of the gamified Katetov order is isomorphic to the original LT-order. This was a surprising discovery, and opens up many challenging questions regarding the interplay between combinatorial and computable complexity, which informed the rest of the paper's investigations. This talk will discuss some of these interactions.

The Skolemization method in first-order logic is a well-known translation on formulas that makes explicit the implicit functions hidden in quantifier combinations ∀∃. It is an important tool in certain areas in computer science but it also has intriguing properties from the viewpoint of logic. For classical logic, Skolemization is complete, meaning that a Skolemized formula is derivable if and only if its original is. But in intuitionistic and various other intermediate logics this is no longer the case.

In this talk I will give a short survey of some results about Skolemization in nonclassical logics. In particular, I will explain the necessary and sufficient condition for the completeness of Skolemization in intermediate logics, thereby showing how principles that express quantifier shifts over connectives determine whether in a given logic Skolemization is complete. The locally unsound proof systems introduced by Aguilera and Baaz play an important role in obtaining these results.

This is joint work with Matthias Baaz, Mariami Gamsakhurdia, and Raheleh Jalali.

In light of the $G_0$-dichotomy (Kechris-Solecki-Todorčević), Lecomte and Zeleny asked for least graphs with no $Σ^0_α$-measurable countable colourings; the provided examples for $α = 1,2,3$. We describe classes of graphs with no $Σ^0_α$ colourings, using notions of forcing inspired by Steel's tagged tree forcing and used by Steel and Slaman when studying arithmetic equivalence as a Borel equivalence relation. These notions of forcing are also useful in calculating the Borel rank of some classes from computability theory (namely, the highα sets), and for a new proof of a non-separation result for the iterated Frechet ideals and filters, due to Debs and Saint Raymond. This is largely joint work with Lecomte, Turetsky, and Zeleny.