---
tag: CMC2C1D53W
kind: concept
name: Woodin cardinal
msc: ["03E55", "03E60"]
specializes: [X1FG7PSCTY]
sources:
  - url: https://en.wikipedia.org/wiki/Woodin_cardinal
same_as:
  wikidata: Q2548234
  wikipedia: Woodin cardinal
---

## Summary

A Woodin cardinal is a [[large cardinal|X1FG7PSCTY]] defined by [[elementary embeddings|CGH8C8EA9N]] that reach high into the universe.
It is named after W. Hugh Woodin.
Woodin cardinals are the large cardinals behind [[determinacy|QXECDPAYTN]]: Martin and Steel proved that infinitely many Woodin cardinals imply [[projective determinacy|2GY3131ZPS]].
A Woodin cardinal is Mahlo, but it need not be weakly compact or measurable.

## Explanations

### L1 human

A Woodin cardinal is a size of infinity below which there are many cardinals that are the starting point of embeddings of the universe into a copy of itself.
These embeddings must keep a large part of the universe intact, and how large is controlled by any given function.
Woodin cardinals are the large cardinals one needs to show that many infinite games on the reals are determined.
So they connect the theory of large cardinals with the study of sets of real numbers.