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