Large cardinals

cardinal number in set theory not provable from ZFC

cardinal unobtainable from smaller cardinals via usual cardinal arithmetic

transitive set closed under the operations of parameterized union, power set and unordered pairs

large cardinal number that is the critical point of a nontrivial elementary embedding of the universe into a transitive class

set-theoretic statement, consistent with ZF but contradicting choice, that any game corresponding to a subset of Baire space is determined

Mahlo cardinal 𝜅 such that every height‐𝜅 tree either has a size‐𝜅 level or a size‐𝜅 branch

set theory construction

inaccessible cardinal number 𝜅 such that the set of inaccessibles less than 𝜅 is stationary in 𝜅