Mathematical logic and foundations Definition

Supercompact cardinal

Builds on Cardinal number Elementary embedding Ordinal number

Definition

Let $\kappa$ be a cardinal, and let $\lambda \ge \kappa$ be an ordinal. The cardinal $\kappa$ is $\lambda$ -supercompact if there is an elementary embedding $j \colon V \to M$ , with $M$ a transitive class, such that:

The critical point of $j$ is $\kappa$ , that is, $j(\alpha) = \alpha$ for $\alpha < \kappa$ and $j(\kappa) > \kappa$ .
$j(\kappa) > \lambda$ .
$M$ is closed under $\lambda$ -sequences: every function from $\lambda$ to $M$ is an element of $M$ .

The cardinal $\kappa$ is supercompact if it is $\lambda$ -supercompact for every ordinal $\lambda \ge \kappa$ .

Remarks

A cardinal is $\kappa$ -supercompact if and only if it is measurable: the ultrapower embedding by a $\kappa$ -complete nonprincipal ultrafilter on $\kappa$ satisfies the three clauses with $\lambda = \kappa$ . So the least measurable cardinal is $\kappa$ -supercompact, but it is not supercompact, since a supercompact cardinal has measurable cardinals below it. An equivalent form uses ultrafilters on the set $P_\kappa(\lambda)$ of subsets of $\lambda$ of size less than $\kappa$ . The cardinal $\kappa$ is $\lambda$ -supercompact if and only if there is a fine normal $\kappa$ -complete ultrafilter on $P_\kappa(\lambda)$ . Every supercompact cardinal is a Woodin cardinal.

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Tag	`SC3MSJRJ3S`
URI	`m2://SC3MSJRJ3S`
Kind	definition
Writing	W0 stub
MSC	03E55
Defines	Supercompact cardinal
Uses	Cardinal number, Elementary embedding, Ordinal number
Changed	2 November 2026

Supercompact cardinal

Definition

Remarks

Reading path

Further reading

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