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 the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. Therefore the discussion of large cardinals takes place in a realm of conditional proofs, which (according to the consensus view of logicians) will remain so. The following is a list of some types of large cardinals; it is arranged in order of the consistency strength. Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for all listed cardinal descriptions φ of lesser consistency strength, V(κ) satisfies "there are unboundedly many cardinals satisfying φ".

weakly inaccessible cardinals
strongly inaccessible cardinals (actually the same consistency strength as weakly inaccessible)
Mahlo cardinals
n-Mahlo cardinals
weakly compact cardinals
Π^m_n-indescribable cardinals
totally indescribable cardinals
unfoldable cardinals
subtle cardinals
ineffable cardinals
remarkable cardinals
0^# (not a cardinal, but proves the existence of transitive models with the cardinals above)
Erdős cardinals (The existence of the Aleph-1-Erdős cardinal implies the existence of 0#, which implies the consistency of Erdős cardinals for all countable ordinals.)
Almost Ramsey cardinals
Ramsey cardinals
measurable cardinals
λ-strong cardinals
strong cardinals
Woodin cardinals
weakly hyper-Woodin cardinals
Shelah cardinals
hyper-Woodin cardinals
superstrong cardinals
strongly compact cardinals
supercompact cardinals
extendible cardinals
huge cardinals
superhuge cardinals
n-huge cardinals
super-n-huge cardinals
rank-into-rank

Other types of large cardinals

λ-unfoldable cardinals
n-ineffable cardinals
totally ineffable cardinals
η-extendible cardinals
almost huge cardinals (between supercompact and huge cardinals)
weakly-ineffable cardinals

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