Home > Large cardinal

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
• totally indescribable cardinals
• unfoldable cardinals
• subtle cardinals
• ineffable cardinals
• remarkable cardinals
• Erdos cardinalsIn mathematics, an Erdos cardinal is a certain kind of large cardinal number. Formally, a cardinal number κ which is the least cardinal such that for every function f κ → {0, 1} there is a set of order type α that is homogeneous for
• 0^#Set theory In mathematical set theory, 0 (zero sharp, also: 0#) is defined to be a particular real number satisfying certain conditions. The definition is a bit awkward, because there may in fact be no real number satisfying the conditions. The propositio (not a cardinal, but proves the existence of transitive models with the cardinals above)
• Ramsey cardinalIn mathematics, a Ramsey cardinal is a certain kind of large cardinal number. Formally, a cardinal number κ such that for every function f κ → {0, 1} (with κ denoting the set of all finite subsets of κ) there is a set of cardis
• measurable cardinalIn mathematics, a measurable cardinal is a certain kind of large cardinal number. Formally, a measurable cardinal is a cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. Equivales
• strong cardinalIn mathematics, a strong cardinal is a cardinal number κ such that for all ordinal numbers λ there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and V ⊆ M''. A lambda;-sts
• Woodin cardinalIn mathematical logic, a Woodin cardinal is a cardinal number κ such that for all f : κ → κ there exists :α < κ with f α] ⊆ α and an elementary embedding j : V → M from V into a transitive inner mods
• weakly hyper-Woodin cardinalIn axiomatic set theory, weakly hyper-Woodin cardinals are a kind of large cardinals. A cardinal κ is called weakly hyper-Woodin iff for every set S there exists a normal measure U on κ such that the set {λ < κ | λ is &kapps
• Shelah cardinalIn axiomatic set theory, Shelah cardinals are a kind of large cardinals. A cardinal κ is called Shelah iff for every f : κ → κ, there exists a transitive class N and a j : V → N with critical point κ and V ⊂ N''.s
• hyper-Woodin cardinals
• superstrong cardinals
• supercompact cardinals
• extendible cardinals
• huge cardinals
• superhuge cardinal s
• n-huge cardinals
• rank-into-rank