Dianthe Basak (Paris)

Continuous Actions of Monoids, a New Perspective on Large Cardinals

The standard "symmetric models" approach to building models of the failure of the axiom of choice relies on the action of non-discrete topological groups on a universe of sets (either sets-with-atoms or a boolean valued model of names). A new approach interprets large cardinal axioms as positing the action of non-discrete topological monoids on the universe of sets (a model of names, a model with atoms, or even $V$ itself). By understanding such actions, one can reinterpret (and reprove) the Kunen inconsistency theorem as a symmetric model theorem, as well as interpreting a version of the HOD conjecture in a natural way. The talk will attempt to place such techniques in their natural context and highlight some applications.