Logic group

In classical set theory, Gödel's constructible universe 𝐿 enjoys strong absoluteness properties and remains unchanged in forcing extensions. However, Heyting-valued forcings portray a very different picture by introducing new non-classical ordinals (i.e. ordinals not linearly ordered by the membership relation), and thus new elements in 𝐿, in intuitionistic extensions that violate the law of excluded middle.

The method of incomparable codings is a family of approaches I developed in my PhD to use such ordinals to control (especially, enlarge) 𝐿. In arXiv:2601.23070 [math.LO], I proved the following theorem: for any set z in a ZFC universe, there is a Heyting-valued extension where its powerset 𝒫(ž) ∈ 𝐿. In this talk, we will provide a sneak peek of this mechanism by building the forcing extension needed for 𝒫(ω) ∈ 𝐿; if time allows, we will briefly talk about technicalities and new developments on how this extends to sets larger than ω.

Henselian valued fields have a central place in model theory: they provide natural examples of tame theories in the sense of classification theory, while also offering a rich setting in which to study complexity phenomena in model theory. In this talk, I will survey some fundamental model-theoretic results on Henselian valued fields and present new results from joint work with Paul Wang, in which we generalize results from the algebraically closed valued field setting to the broader Henselian context.

NOTES: Unusual time 16:00. Part of workshop on "Tame Geometry and Combinatorics".
Large fields are an interesting class of fields first introduced by Pop in the 90's for Galois-theoretic reasons. They have subsequently been studied for a variety of reasons. As logicians we are interested in large fields because essentially all known logically tame fields are large. I will discuss recent work on the model theory of large fields, joint with Will Johnson, Chieu-Minh Tran, and Jinhe Ye. Only minimal background in algebra will be assumed.

Ultraproducts of rings and their modules have been appearing in various places in representation theory and the theory of tensor categories.  We consider how this is reflected in the model theory of modules over the rings and, in particular, we describe the effect on the lattices of pp formulas, definable subcategories, Ziegler spectra and the associated abelian categories of pp-imaginaries.  We also look at what happens when we enrich the picture with a monoidal = tensor product structure.

The theory of ZFC without powerset is interpretable in the system $Z_2$ of second order arithmetic. The interpretation is usually done in two steps. The first is to interpret an intermediate theory of sets by considering well founded trees modulo an appropriate equivalence relation. The second is by defining the constructible universe $L$ in the intermediate system.

I will go over some of the ways such interpretations are suited in more general systems that may have uncountable sets and restricted separation. I will also talk about how it turns out that the weakest principle needed to carry out the tree interpretation can be characterized in terms of clopen games and transfinite recursion. This is a joint work with Emanuele Frittaion.