Fudan Logic Seminar 2025

October 17

Location: TBA

Speaker: 徐明

Time: 13:30 - 15:10

FMP/RA/FA-Ensuring Logics: Their Boundaries and Relations

For each normal modal logic L and each property P, we call L P-ensuring if all normal extensions of L have the property P. Let FMP, RA, FA and COMP stand respectively for the finite model property, recursive axiomatizability, finite axiomatizability and (Kripke) completeness. Many classical results in the literature of modal logic establish certain logics to be FMP-ensuring, FA-ensuring or COMP-ensuring. We naturally wonder whether there is a syntactical way to capture all FMP-ensuring logics or all FA-ensuring logics etc. in a well-known class of modal logics, and how these types of logics are (extensionally or set-theoretically) related within the class. NExtK4.3 is a relatively small class, despite the fact that it is a continuum. Yet all members of it are COMP-ensuring by Fine's theorem of completeness for transitive logics of finite width, and many members of it are known to be FMP-ensuring and FA-ensuring, which makes it a good place to start a study of such logics. We will present a necessary and sufficient condition for logics in NExtK4.3 to be FMP-ensuring, another such condition for them to be RA-ensuring, and another such condition for them to be FA-ensuring. These conditions enable us to draw many interesting conclusions, including that all FMP-ensuring logics in Next4.3 are FA-ensuring.

July 30

Location: HGX208, Tencent meeting ID: 206152768

Speaker: Alf Onshuus

Time: 9:30 - 10:30

On the limit theory of numerical semigroups with two generators

This talk will be about the limit theory of numerical semigroups, namely, cofinal submonoids of the natural numbers. The basic example is the following: Let $a$ and $b$ be relatively prime natural numbers, and consider all linear combinations of $a$ and $b$ with non-negative integer coefficients. It is an elementary (and well known) fact to show that these are all cofinite. These are precisely the numerical semigroups with two generators.
The talk will concentrate on the theory of ultraproducts of numerical semigroups with two generators. We will explain why we believe that this theory is definable, and exhibit a complete axiomatization in some particular cases.

July 23

Location: HGX208

Speaker: Joel David Hamkins

Time: 14:30 - 15:30

Computable surreal numbers

I shall give an account of the theory of computable surreal numbers, proving that these form a real-closed field. Which real numbers arise as computable surreal numbers? You may be surprised to learn that some noncomputable real numbers are computable as surreal numbers, and indeed the computable surreal real numbers are exactly the hyperarithmetic reals. More generally, the computable surreal numbers are exactly those with a hyperarithmetic surreal sign sequence. This is joint work with Dan Turetsky, but we subsequently found that it is a rediscovery of earlier work of Jacob Lurie.