Large Cardinals in Weakened Axiomatic Theories

We study the notion of non-trivial elementary embeddings j from the set-theoretic universe, V, to itself under the assumption that V satisfies various classical and intuitionistic set theories. In particular, we investigate what consequences can be derived if V is only assumed to satisfy Kripke Platek set theory, set theory without Power Set or intuitionistic set theory.

To do this, we construct the constructible universe in Intuitionistic Kripke Platek without Infinity and use this to find lower bounds for such embeddings. We then study the notion of definable embeddings before giving some initial bounds in terms of the standard large cardinal hierarchy. Finally, we give sufficient requirements for there to be no non-trivial elementary embedding of the universe in ZFC without Power Set.

As a by-product of this analysis, we also study Collection Principles in ZFC without Power Set. This leads to models witnessing the failure of various Dependent Choice Principles and to the development of the theory of the Respected Model, a generalisation of symmetric submodels to the class forcing context.