Richard Matthews (Université Paris-Est Créteil)

Very large set axioms over Constructive Set Theories

One of the main areas of research in set theory is the study of large cardinal axioms and many of these can be characterised by the existence of elementary embeddings with certain properties. The guiding principle is then that the closer the domain and co-domain of the embedding is to the universe, the stronger the resulting large cardinal axiom. This leads naturally to the question of whether there is an elementary embedding of the universe into itself which is not the identity, and the least ordinal moved by such an embedding is known as a Reinhardt cardinal. While Kunen famously proved that no such embedding can exist if the universe satisfies ZFC, it is an open question in many subtheories of ZFC, most notably ZF (without Choice).

In this talk we will study elementary embeddings in the weaker context of intuitionistic set theories, that is set theories without the law of excluded middle. We shall observe that the ordinals can be very ill-behaved in this setting and therefore we will reformulate large cardinals by instead looking for large sets which capture the desired structural properties. We shall investigate the consistency strength of analogues to measurable cardinals, Reinhardt cardinals and many other similar ideas in terms of the standard ZFC large cardinal hierarchy.

This is joint work with Hanul Jeon.