# List of large cardinal properties

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This page includes a list of cardinals with large cardinal properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, Vκ satisfies "there is an unbounded class of cardinals satisfying φ".

The following table usually arranges cardinals in order of consistency strength, with size of the cardinal used as a tiebreaker. In a few cases (such as strongly compact cardinals) the exact consistency strength is not known and the table uses the current best guess.

Finally, if there were a nontrivial elementary embedding from the entire von Neumann universe V into itself, j:VV, its critical point would be called a Reinhardt cardinal. It is provable in ZFC that there is no such embedding (and therefore no Reinhardt cardinals). However their existence has not yet been refuted in ZF alone (that is, without use of the axiom of choice).

## References

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