Small embeddings, forcing with side conditions, and large cardinal characterizations

Abstract

In this thesis, we provide new characterizations for several well-studied large cardinal notions. These characterizations will be of two types. Motivated by seminal work of Magidor, the first type characterizes large cardinals through the existence of so-called small embeddings, elementary embeddings between set-sized structures that map their critical point to the large cardinal in question. Building up on these characterizations, we also provide characterizations of many large cardinal notions through the validity of certain combinatorial principles at omega_2 in generic extensions. The combinatorial principles used in these characterizations are generalizations of large cardinal properties defined through small embeddings that can also hold at accessible cardinals and, for inaccessible cardinals, these principles are equivalent to the original large cardinal property. In this thesis, we focus on generic extensions obtained via the pure side condition forcing introduces by Neeman in his studies of forcing axioms and their generalizations. <br /> Our results will provide these two types of characterizations for some of the most prominent large cardinal notions, including inaccessible, Mahlo, Pi^m_n-indescribable cardinals, subtle, lambda-ineffable, and supercompact cardinals. In addition, we will derive small embedding characterizations of measurable, lambda-supercompact and huge cardinals, as well as forcing characterizations of almost huge and super almost huge cardinals. As an application of techniques developed in this work, we provide new proofs of Weiß 's results on the consistency strength of generalized tree properties, eliminating problematic arguments contained in his original proofs.<br /> The work presented in this thesis is joint work with Peter Holy and Philipp Lücke. It will be published in the following papers:<br /> Peter Holy, Philipp Lücke and Ana Njegomir. Small Embedding Characterizations for Large Cardinals. Annals of Pure and Applied Logic. Volume 170, Issue 2, pp. 251-271, 2019.<br /> Peter Holy, Philipp Lücke and Ana Njegomir. Characterizing large cardinals through Neeman's pure side condition forcing. Submitted to Fundamenta Mathematicae, 28 pages, 2018.