Cardinals in Isabelle/HOL

We report on a formalization of ordinals and cardinals in Isabelle/HOL. A main challenge we faced was the inability of higher-order logic to represent ordinals canonically, as transitive sets (as done in set theory). We resolved this into a “decentralized” representation identifying ordinals with well-orders, with all concepts and results proved to be invariant under order isomorphism. We also discuss several applications of this general theory in formal developments

Proof theory of weak compactness

We show that the existence of a weakly compact cardinal over the Zermelo-Fraenkel's set theory is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations

Splitting stationary sets from weak forms of Choice

Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below $\lambda$ of cofinality $\theta$ into $\lambda$ many stationary sets, where $\theta < \lambda$ are regular cardinals. This is a continuation of \cite{Sh835}