Large sets in constructive set theory

This thesis presents an investigation into large sets and large set axioms in the context of the constructive set theory CZF.
We determine the structure of large sets by classifying their von Neumann stages and use a new modified cumulative hierarchy to characterise their arrangement in the set theoretic universe. We prove that large set axioms have good metamathematical properties, including absoluteness for the common relative model constructions of CZF and a preservation of the witness existence properties CZF enjoys. Furthermore, we use realizability to establish new results about the relative consistency of a plurality of inaccessibles versus the existence of just one inaccessible. Developing a constructive theory of clubs, we present a characterisation theorem for Mahlo sets connecting classical and constructive approaches to Mahloness and determine the amount of induction contained in the assertion of a Mahlo set. We then present a characterisation theorem for 2-strong sets which proves them to be equivalent to a logically simpler concept.
We also investigate several topics connected to elementary embeddings of the set theoretic universe into a transitive class model of CZF, where considering different equivalent classical formulations results in a rich and interconnected spectrum of measurability for the constructive case. We pay particular attention to the question of cofinality of elementary embeddings, achieving both very strong cofinality properties in the case of Reinhardt embeddings and constructing models of the failure of cofinality in the case of ordinary measurable embeddings, some of which require only surprisingly low conditions. We close with an investigation of constructive principles incompatible with elementary embeddings.