“For a large mathematical object \(A\) and a property \(P\), if all small sub-objects of \(A\) satisfy \(P\), then \(A\) satisfies \(P\).”

Most important example is the Compactness Theorem in mathematical logic, which states that for any infinite first order theory \(T\), if all finite sub-theories of \(T\) have models, then \(T\) has a model. From this theorem, many compactness statements follow.

In the Compactness Theorem and its corollaries, large objects are infinite ones, and small objects are finite ones. In other words, large ones are of cardinality \(\geq \aleph_0\), and small ones are of cardinality \(< \aleph_0\). Set theorists are interested in compactness statements in which large ones are of cardinality \(\geq \kappa\), and small ones are of cardinality \(< \kappa\), where \(\kappa\) is an uncountable cardinal.

I will talk about these compactness statements at uncountable cardinals. I will introduce classical results and my recent results.

In this talk, we will first give a brief introduction to the modal logic \(\mathbf{K}\), then introduce the pure logic of necessitation \(\mathbf{N}\) that is obtained by removing K from \(\mathbf{K}\), and then discuss the properties of \(\mathbf{N}\) and some of its extensions, including the finite frame property.

In this talk, we will combine these modal logic, and consider the relation between this modal logic and set theory.

This talk will begin with an introduction to cardinal invariants. Then, as an application, we will look at the consistency proof of the Borel conjecture. Finally, the speaker will present a problem he wants to solve related to the Borel conjecture.

In this talk, I will present our attempt to rediscover the proofs of Cobham's and Vaught's theorems and to clarify the relationship between Cobham's theorem and Vaught's theorem from the viewpoint of Pour-El's theorem. This is a joint work with Albert Visser (Utrecht University).

In this talk, we focus on generalizations for singular \(\mu\). In particular, we consider the characteristic \(\mathfrak{d}_{\mu}\), generalized the dominating number \(\mathfrak{d}\). \(\mathfrak{d}_{\mu}\) has an upper bound \(2^\mu\), and the previous research by Shelah gives a sufficient condition for \(\mathfrak{d}_{\mu} = 2^\mu\). The speaker give other sufficient conditions for ``\(\mathfrak{d}_{\mu}\) is large" (i.e. \(\mathfrak{d}_{\mu} = 2^\mu\) or \(\mathfrak{d}_{\mu} \geq 2^{<\mu}\)) when \(\mu = \aleph_\omega\).

Historically, work on formalizing provability predicates of arithmetic in terms of modality goes back to Gödel. His early work focused on the modal logic S4, since it was expected to give a provability semantics of intuitionistic propositional logic. In 1976, Solovay proved that the modal logic GL properly captures the notion of provability predicates of Peano Arithmetic PA.

The connection between S4 and arithmetic has been studied for decades. In 1995 Artemov introduced an explicit version of the modal logic S4, called ``the logic of proofs"(LP). The logic LP treats formulas with ``proof terms", called LP-formulas. He showed two important results: (i) The logic LP is arithmetical complete with respect to proof predicates. That is, LP can be embedded into PA; (ii) Every modal formula provable in S4 can be realized into an LP-formula provable in LP.

In recent years, various sublogics and extensions of LP, so-called justification logics, have been studied. For instance, The logic JT45 is a justification logic counterpart of S5 which is an extension of S4. At the end of this talk, we discuss the arithmetical aspect of JT45.