large cardinal
cardinal number in set theory not provable from ZFC
Woodin cardinal
kind of large cardinal number
weakly inaccessible cardinal
regular weak limit cardinal
reflecting cardinal
large cardinal number 𝜅 for which there is a normal ideal I on 𝜅 such that for every 𝑋∈𝐼⁺, the set of 𝛼∈𝜅 for which 𝑋 reflects at 𝛼 is in 𝐼⁺
Reinhardt cardinal
large cardinal inconsistent with the axiom of choice; the critical point of a nontrivial elementary embedding of the universe into itself
iterable cardinal
kind of large cardinal number
subtle cardinal
kind of large cardinal number
subcompact cardinal
kind of large cardinal number
worldly cardinal
large cardinal number 𝜅 such that the sets of rank <𝜅 is a model of Zermelo–Fraenkel set theory
remarkable cardinal
kind of large cardinal number
tall cardinal
large cardinal number 𝜅 such that, for any ordinal 𝜆, there is an elementary embedding 𝑗: 𝑉→𝑀 with critical point 𝜅 such that 𝑗(𝜅)>𝜆 and 𝑀 contains all 𝜅‐sequences of 𝑀
ineffable cardinal
kind of large cardinal number
rank-into-rank
large cardinal property given by elementary embeddings of initial fragments of the von Neumann hierarchy V
indescribable cardinal
kind of large cardinal number
Berkeley cardinal
large cardinal number 𝜅, incompatible with the axiom of choice, such that for any transitive set 𝑀∋𝜅 and 𝛼<𝜅, there is a nontrivial elementary embedding 𝑀→𝑀 with critical point strictly between 𝛼 and 𝜅