Erdős cardinal
In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by.
The Erdős cardinal is defined to be the least cardinal such that for every function there is a set of order type that is homogeneous for . In the notation of the partition calculus, the Erdős cardinal is the smallest cardinal such that
Existence of zero sharp implies that the constructible universe satisfies "for every countable ordinal, there is an -Erdős cardinal". In fact, for every indiscernible satisfies "for every ordinal, there is an -Erdős cardinal in ".
However, existence of an -Erdős cardinal implies existence of zero sharp. If is the satisfaction relation for , then existence of zero sharp is equivalent to there being an -Erdős ordinal with respect to. And this in turn, the zero sharp implies the falsity of axiom of constructibility, of Kurt Gödel.
If κ is -Erdős, then it is -Erdős in every transitive model satisfying " is countable".
