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Concept
Erdős cardinal
Summary
In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .
A cardinal κ is called α-Erdős if for every function f : κ< ω → {0, 1}, there is a set of order type α that is homogeneous for f . In the notation of the partition calculus, κ is α-Erdős if
κ(α) → (α)< ω.
The 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, the existence of an ω1-Erdős cardinal implies existence of zero sharp. If f is the satisfaction relation for L (using ordinal parameters), then the existence of zero sharp is equivalent to there being an ω1-Erdős ordinal with respect to f . Thus, the existence of zero sharp implies that the axiom of constructibility is false.
If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable.
Official source
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