Английская Википедия:Erdős cardinal
In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Шаблон:Harvs.
A cardinal κ is called α-Erdős if for every function Шаблон:Math there is a set of order type Шаблон:Mvar that is homogeneous for Шаблон:Math. In the notation of the partition calculus, κ is α-Erdős if
The existence of zero sharp implies that the constructible universe Шаблон:Mvar satisfies "for every countable ordinal Шаблон:Mvar, there is an Шаблон:Mvar-Erdős cardinal". In fact, for every indiscernible Шаблон:Mvar satisfies "for every ordinal Шаблон:Mvar, there is an Шаблон:Mvar-Erdős cardinal in Шаблон:Math" (the Levy collapse to make Шаблон:Mvar countable).
However, the existence of an Шаблон:Math-Erdős cardinal implies existence of zero sharp. If Шаблон:Math is the satisfaction relation for Шаблон:Mvar (using ordinal parameters), then the existence of zero sharp is equivalent to there being an Шаблон:Math-Erdős ordinal with respect to Шаблон:Math. Thus, the existence of zero sharp implies that the axiom of constructibility is false.
If κ is Шаблон:Mvar-Erdős, then it is Шаблон:Mvar-Erdős in every transitive model satisfying "Шаблон:Mvar is countable."
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