Critical Cardinals

TL;DR

This paper introduces the concept of critical cardinals, explores their properties, and demonstrates their consistency relative to supercompact cardinals, revealing new insights into large cardinal theory.

Contribution

It defines critical cardinals independently of the axiom of choice and establishes their consistency relative to supercompact cardinals.

Findings

Critical cardinals are distinct from measurable cardinals without choice.

It is consistent that a critical cardinal's successor can be singular.

A technical criterion for lifting elementary embeddings is developed.

Abstract

We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the axiom of choice this is equivalent to measurability, but it is well-known that choice is necessary for the equivalence. Oddly enough, this central notion was never investigated on its own before. We prove a technical criterion for lifting elementary embeddings to symmetric extensions, and we use this to show that it is consistent relative to a supercompact cardinal that there is a critical cardinal whose successor is singular.