---
tag: 91GE4C1BC5
kind: concept
name: Supercompact cardinal
msc: ["03E55"]
specializes: [AKBWT323RQ]
sources:
  - url: https://en.wikipedia.org/wiki/Supercompact_cardinal
same_as:
  wikidata: Q7643154
  wikipedia: Supercompact cardinal
---

## Summary

A supercompact cardinal is a cardinal $\kappa$ that is the critical point of many [[elementary embeddings|CGH8C8EA9N]] $j \colon V \to M$.
For each length, one of them has a target $M$ closed under sequences of that length.
It strengthens the embedding form of [[measurability|AKBWT323RQ]], which asks only for closure under sequences of length $\kappa$.
Supercompact cardinals are [[Woodin cardinals|CMC2C1D53W]] and lie high in the hierarchy of [[large cardinals|X1FG7PSCTY]].
They are used to prove the consistency of strong axioms such as the [[proper forcing axiom|35XTXNEWQ9]].

## Explanations

### L1 human

A measurable cardinal gives an embedding of the universe into a copy of itself that keeps sequences of length $\kappa$.
A supercompact cardinal gives such embeddings that keep sequences of every length, as long as one fixes the length in advance.
So the copy looks more and more like the whole universe.
This makes supercompact cardinals very strong, and many consistency results use them.