- Erdős cardinal
In

mathematics , an**Erdős cardinal**(named afterPaul Erdős ) is a certain kind oflarge cardinal number.Formally, a

cardinal number κ which is the least cardinal such that for every function "f": κ^{ < ω }→ {0, 1} there is a set of order type α that is homogeneous for "f", is called an**α-Erdős**cardinal.Existence of

zero sharp implies that the constructible universe L satisfies "for every countable ordinal α, there is an α-Erdős cardinal". In fact, for every indiscernible κ, L_{κ}satisfies "for every ordinal α, there is an α-Erdős cardinal in Coll(ω, α) (the generic collapse to make α countable)".However, existence of an ω

_{1}-Erdős cardinal implies existence of zero sharp. If f is the satisfaction relation for L (using ordinal parameters), then existence of zero sharp is equivalent to there being an ω_{1}-Erdős ordinal with respect to f.If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable".

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