The Ultrapower Axiom

TL;DR

This paper introduces the Ultrapower Axiom, a principle expected to hold in canonical models of set theory, and explores its implications for the inner model problem and the relationship between strong compactness and supercompactness.

Contribution

It proposes the Ultrapower Axiom as a new principle and investigates its consequences, providing evidence toward solving the inner model problem for supercompact cardinals.

Findings

Under the Ultrapower Axiom, strong compactness and supercompactness are equivalent.

The Ultrapower Axiom offers a new framework to approach the inner model problem.

Evidence suggests the inner model problem may be solvable using this principle.

Abstract

The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The problem is closely related to the more precise question of the equiconsistency of strongly compact cardinals and supercompact cardinals. This dissertation approaches these two problems abstractly by introducing a principle called the Ultrapower Axiom which is expected to hold in all known canonical models of set theory. By investigating the consequences of the Ultrapower Axiom under the hypothesis that there is a supercompact cardinal, we provide evidence that the inner model problem can be solved. Moreover, we establish that under the Ultrapower Axiom, strong compactness and supercompactness are essentially equivalent.