Erdős cardinal
In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by.
A cardinal is called -Erdős if for every function, there is a set of order type that is homogeneous for. In the notation of the partition calculus, is -Erdős if
Under this definition, any cardinal larger than the least -Erdős cardinal is -Erdős.
The 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, the existence of an -Erdős cardinal implies existence of zero sharp. If is the satisfaction relation for, then the existence of zero sharp is equivalent to there being an -Erdős ordinal with respect to. Thus, the existence of an -Erdős cardinal implies that the axiom of constructibility is false.
The least -Erdős cardinal is not weakly compact,p. 39. nor is the least -Erdős cardinal.p. 39
If is -Erdős, then it is -Erdős in every transitive model satisfying " is countable."
Dodd's Notion of Erdős Cardinals
For a limit ordinal, a cardinal is less often called -Erdős if for every closed unbounded and every function such that for all, there is a set of order-type that is homogeneous for.p. 138.An equivalent definition is that is -Erdős if for any, there is a set of order-type of order-indiscernibles for the structure such that:
- for every,, and
- for every, the set forms a set of order-indiscernibles for the structure.