Large cardinal
In
mathematics, a
cardinal is called a
large cardinal if it belongs to a class of cardinals the existence of which (provably) cannot be proved within
ZFC (assuming ZFC itself is consistent). Here are some large cardinals, arranged in order of the consistency strength:
- weakly inaccessible cardinals
- strongly inaccessible cardinals (actually the same consistency strength as weakly inaccessible)
- Mahlo cardinals
- n-Mahlo cardinals
- weakly compact cardinals
- totally indescribable cardinals
- subtle cardinals
- ineffable cardinals
- remarkable cardinals
- 0^{#} (not a cardinal, but proves the existence of transitive models with the cardinals above)
- Ramsey cardinals
- measurable cardinals
- strong cardinals
- Woodin cardinals
- Shelah cardinals
- superstrong cardinals
- supercompact cardinals
- extendible cardinals
- huge cardinals
- n-huge cardinalss
- rank-into-rank
and some other cardinals (unordered):
- unfoldable cardinals
- Erdös cardinals
- super-almost-huge cardinals
- superhuge cardinals