Large cardinals - Definition

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
• supercompact cardinals
• extendible cardinals
• huge cardinals
• superhuge cardinals
• n-huge cardinals
• rank-into-rank

Other types of large cardinals

• λ-unfoldable cardinals
• n-ineffable cardinals
• totally ineffable cardinals
• η-extendible cardinals