Erdős cardinal

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In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958).

The Erdős cardinal κ(α) is defined to be 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 (if such a cardinal exists). In the notation of the partition calculus, the Erdős cardinal κ(α) is the smallest cardinal such that

κ(α) → (α)< ω

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, Lκ satisfies "for every ordinal α, there is an α-Erdős cardinal in Coll(ω, α) (the Levy collapse to make α countable)".

However, existence of an ω1-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 ω1-Erdős ordinal with respect to f. And this in turn, the zero sharp implies the falsity of axiom of constructibility, of Kurt Gödel.

If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable".

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