Higher Structural Reflection and Very Large Cardinals

Abstract

One line of research in set theory aims at deriving large cardinal axioms from strengthened forms of reflection principles. This research is often motivated by the foundational goal of justifying the large cardinal axioms. The most comprehensive attempt in this direction is the program of structural reflection (SR), initiated by Joan Bagaria, whose ultimate goal is to formulate all large cardinal axioms as instances of a single, general structural reflection principle that is conceptually compelling. The basic version of SR already gives the hierarchy of large cardinals from supercompact cardinals, through C(n)-extendible cardinals, up to Vopěnka’s Principle. A stronger version of SR, the exact structural reflection principle (ESR), is studied by Bagaria and Philipp Lücke, which gives almost huge cardinals, and beyond. However, ESR differs in form from the basic version of SR, rather than being direct generalization of the same principle. In this thesis we formulate the level by level version and the capturing version of SR (CSR). CSR is a direct generalization of the basic version of SR. We introduce and study the m-supercompact cardinals, the C(n)-m-fold extendible cardinals, and the capturing version of VP, and show that the pattern of correspondence between large cardinals and the basic version of SR also extends to the higher realm. We also apply our results to answer several open questions concerning ESR. Finally, we note that CSR, when generalized to its ω-version, leads to inconsistency.