Large Cardinals

Large cardinals are strong axioms of infinity asserting the existence of cardinals so large that their existence cannot be proved in ZFC, and they form a nearly linear hierarchy that calibrates the strength of mathematical theories.

Definition

A large cardinal axiom asserts the existence of a cardinal with a strong closure or reflection property, typically expressible through an elementary embedding of the universe; such cardinals exceed what ZFC can prove to exist and so increase the consistency strength of the theory.

Scope

This topic covers the principal large cardinal notions such as inaccessible, Mahlo, weakly compact, measurable, and supercompact cardinals, their characterizations via reflection and elementary embeddings, the consistency-strength hierarchy they generate, and their connections to determinacy and inner model theory.

Core questions

• What closure and reflection properties define the main large cardinals?
• How do elementary embeddings characterize measurable and stronger cardinals?
• Why do large cardinals form an almost linear hierarchy of consistency strength?
• How do large cardinals interact with determinacy and the structure of the reals?

Key theories

Inaccessible and Mahlo cardinals

An inaccessible cardinal is regular and a strong limit, so it cannot be reached by the usual set operations and gives a natural model of ZFC; Mahlo cardinals reflect inaccessibility, beginning the hierarchy.

Measurable cardinals and elementary embeddings

A measurable cardinal carries a nontrivial countably complete ultrafilter, equivalently it is the critical point of an elementary embedding of the universe into an inner model, contradicting the axiom of constructibility.

Consistency-strength hierarchy

Large cardinal axioms are ordered by relative consistency, so that the consistency of one implies that of all weaker ones, providing a yardstick against which the strength of arbitrary theories is measured.

Clinical relevance

Large cardinals supply the canonical scale of consistency strength in mathematics: many statements turn out to be equiconsistent with the existence of some large cardinal, and strong large cardinals imply regularity properties of the real line such as projective determinacy and Lebesgue measurability of definable sets.

History

Inaccessible cardinals arose from Zermelo's and Sierpinski-Tarski's study of models of set theory, and Ulam's 1930 work on measure led to measurable cardinals. Scott showed in 1961 that a measurable cardinal refutes the axiom of constructibility, and the subsequent work of Solovay, Martin, Woodin, and others built the modern hierarchy and its links to determinacy.

Key figures

• Stanislaw Ulam
• Dana Scott
• Robert Solovay
• Hugh Woodin

Seminal works

• kanamori2009
• jech2003
• kunen2011

Frequently asked questions

Why can ZFC not prove that large cardinals exist?

An inaccessible cardinal yields a set model of ZFC, so by Goedel's second incompleteness theorem ZFC cannot prove that such a cardinal exists without proving its own consistency, which it cannot do. The same reasoning applies, a fortiori, to stronger large cardinals.

Why study axioms that cannot be proved consistent?

Large cardinals provide a coherent and well-ordered scale for comparing the strength of mathematical theories, and they settle otherwise independent questions about definable sets of reals, making them a central organizing tool even though their consistency must be assumed.