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 " (the Lévy collapse to make countable).
However, the existence of an -Erdős cardinal implies existence of zero sharp. If is the satisfaction relation for (using ordinal parameters), 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,F. Rowbottom, "Some strong axioms of infinity incompatible with the axiom of constructibility". Annals of Mathematical Logic vol. 3, no. 1 (1971).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 .A. J. Dodd (1982), The Core Model. Cambridge University Press. 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 .
The least cardinal to satisfy the partition relation is still -Erdős under this definition. Every -Erdős cardinal is an inaccessible limit of ineffable cardinals.
Strength relative to other large cardinals For any ordinal α, α-Erdős cardinal is stronger than α-iterable cardinal, but weaker than α+1-iterable cardinal. virtually rank-into-rank cardinal is weaker than ω-Erdős cardinal. If α < ω1, then α-Erdős cardinal is weaker than Silver cardinal. ω1-Erdős cardinal is stronger than zero sharp.
See also List of large cardinal properties
References
Citations