The Landscape of Large Cardinals

TL;DR

This paper provides an introductory overview of the hierarchy of large cardinals in set theory, explaining their significance, definitions, and connections to foundational questions like the Continuum Hypothesis.

Contribution

It offers a clear, accessible survey of large cardinal concepts, their motivations, and their role in understanding the higher infinite in set theory.

Findings

Clarifies the hierarchy of large cardinals

Explores connections with the Continuum Hypothesis

Provides philosophical reflections on consistency beliefs

Abstract

The purpose of this paper is to provide an introductory overview of the large cardinal hierarchy in set theory. By a large cardinal, we mean any cardinal $\kappa$ whose existence is strong enough of an assumption to prove the consistency of ZFC. We assume basic familiarity with set theory, model theory, and the ZFC axioms, though certain concepts will be reviewed as necessary. We attempt to clarify the vast landscape of the higher infinite, motivating the definitions of some (but certainly not all) of the known large cardinals. We also discuss connections with the Continuum Hypothesis and provide some philosophical reflections on belief in the consistency of large cardinals.