In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provablycannot be proved within the standard axiomatic set theoryZFC, 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".
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.)
In mathematics, a cardinal is called a largecardinal 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 largecardinals takes place in a realm of conditional proofs, which (according to the consensus view of logicians) will remain so.
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.)
In mathematical set theory, a strong cardinal or λ-strong cardinal is a type of largecardinal.
Specifically, a λ-strong cardinal is a cardinal number κ such that exists an elementary embedding j from V into a transitive inner model M with critical point κ and
It should be noted that the least strong cardinal is larger than the least Woodin, superstrong, etc. cardinals, but that the consistency strength of strong cardinals is lower: For example, if κ is Woodin, then V
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