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