Strong Hypotheses

Abstract

This chapter provides the basic theory for the strong hypotheses that generalize measurability. §22 discusses Solovay and Reinhardt’s concept of supercompactness, a global reflection property, and the relation of supercompactness to strong compactness. §23 describes the stronger hypotheses that evolved from Reinhardt’s proposals: extendibility and a prima facie extension shown inconsistent by Kunen. §24 then considers hypotheses on the verge of that inconsistency, and then spanning the expanse, n-hugeness and Vopěnka’s Principle. Pursuing offshoots of the theory of supercompactness, §25 describes the combinatorial study of ℘ _κ γ , and §26 provides the fundamentals of extenders and related large cardinals, refined concepts that were to lead to major advances in inner model theory.