Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

I0 and rank-into-rank axioms

This is a survey about I0 and rank-into-rank axioms, with some previously unpublished proofs

Forcing with Non-wellfounded Models

We develop the machinery for performing forcing over an arbitrary (possibly non-wellfounded) model of set theory. For consistency results, this machinery is unnecessary since such results can always be legitimately obtained by assuming that the ground model is (countable) transitive. However, for establishing properties of a given (possibly non-wellfounded) model, the fully developed machinery of forcing as a means to produce new related models can be useful. We develop forcing through iterated forcing, paralleling the standard steps of presentation found in [19] and [14]

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

Forcing with Non-wellfounded Models

We develop the machinery for performing forcing over an arbitrary (possibly non-wellfounded) model of set theory. For consistency results, this machinery is unnecessary since such results can always be legitimately obtained by assuming that the ground model is (countable) transitive. However, for establishing properties of a given (possibly non-wellfounded) model, the fully developed machinery of forcing as a means to produce new related models can be useful. We develop forcing through iterated forcing, paralleling the standard steps of presentation found in [19] and [14]