nLab large cardinal





The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



A large cardinal is a cardinal number that is larger than can be proven to exist in the ambient set theory, usually ZF or ZFC. Large cardinals arrange themselves naturally into a more or less linear order of size and consistency strength, and provide a convenient yardstick to measure the consistency strength of various other assertions that are unprovable from ZFC.

Set theorists often adopt the existence of certain large cardinals as axioms in the foundation of mathematics.

List of large cardinal conditions

Here is a diagram showing the relation between these:

In the context of ZFC, certain axioms are inconsistent with large cardinal axioms:

ZFC large cardinals consistency strength


A general axiomatic framework for large cardinal axioms is proposed in

  • Arthur Apter, Carlos Diprisco, James Henle, William Swicker, Filter spaces: towards a unified theory of large cardinal and embedding axioms, Annals of Pure and Applied Logic Volume 41, Issue 2, 6 February 1989, Pages 93–106

  • Arthur Apter, Carlos Diprisco, James Henle, William Swicker, Filter spaces. II. Limit ultraproducts and iterated embeddings, Acta Cient. Venezolana 40 (1989), no. 5-6, 311–318.

Last revised on November 13, 2023 at 03:09:37. See the history of this page for a list of all contributions to it.