Christian Espíndola: Achieving completeness: from constructive set theory to large cardinals

Abstract:
This thesis is an exploration of several completeness phenomena, both in the constructive and the classical settings.

The constructive part contains a categorical formulation of several constructive completeness theorems available in the literature, but presented here in an unified framework. We develop them within a constructive reverse mathematical viewpoint, highlighting the metatheory used in each case and the strength of the corresponding completeness theorems.

The classical part of the thesis focuses on infinitary intuitionistic propositional and predicate logic. We consider a first-order axiomatic system with a special rule, transfinite transitivity, that embodies both classical distributivity as well as a form of dependent choice, and prove a completeness theorem in terms of an infinitary Kripke semantics, introducing weakly compact cardinals as the adequate metatheoretical assumption for this development. This clarifies, in categorical terms the classical relation between the model theory of infinitary logics and large cardinal assumptions.