Abstract. In the world of infinite cardinals, combinatorial properties of singular
cardinals are somewhat special. This is especially visible by the fact that
they often exhibit a compactness behaviour, such as in Shelah's singular
compactness theorem or pcf theorem. The cardinal \(\aleph_0\) is also very
special, often because of the compactness. An important example is the
compactness of the first order logic. Therefore it is natural to ask if
there is a compact logic associated to singular cardinals, a question that
we explore. This talk reports on results in progress obtained jointly with
Jouko Väänänen.

Abstract. We investigate the senses in which settheoretic forcing can be
seen as a computational process on the models of set theory. Given an oracle
for the atomic or elementary diagram of a model of set theory \(\langle
M,\in^M\rangle\), for example, we explain senses in which one may compute
\(M\)generic filters \(G\subset P\in M\) and the corresponding forcing
extensions \(M[G]\). Meanwhile, no such computational process is functorial,
for there must always be isomorphic alternative presentations of the same
model of set theory \(M\) that lead by the computational process to
nonisomorphic forcing extensions \(M[G]\not\cong M[G']\). Indeed, there is no
Borel function providing generic filters that is functorial in this sense.
This is joint work with Russell Miller and Kameryn Williams.
