# Erdős cardinal

In mathematics, an **Erdős cardinal**, also called a **partition cardinal** is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958).

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

- .

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,^{[1]}^{p. 39.} nor is the least -Erdős cardinal.^{[1]}^{p. 39}

If is -Erdős, then it is -Erdős in every transitive model satisfying " is countable."

## See also

[edit]## References

[edit]- Baumgartner, James E.; Galvin, Fred (1978). "Generalized Erdős cardinals and 0
^{#}".*Annals of Mathematical Logic*.**15**(3): 289–313. doi:10.1016/0003-4843(78)90012-8. ISSN 0003-4843. MR 0528659. - Drake, F. R. (1974).
*Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76)*. Elsevier Science Ltd. ISBN 0-444-10535-2. - Erdős, Paul; Hajnal, András (1958). "On the structure of set-mappings".
*Acta Mathematica Academiae Scientiarum Hungaricae*.**9**(1–2): 111–131. doi:10.1007/BF02023868. ISSN 0001-5954. MR 0095124. S2CID 18976050. - Kanamori, Akihiro (2003).
*The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings*(2nd ed.). Springer. ISBN 3-540-00384-3.

### Citations

[edit]- ^
^{a}^{b}F. Rowbottom, "Some strong axioms of infinity incompatible with the axiom of constructibility". Annals of Mathematical Logic vol. 3, no. 1 (1971).