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)
