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 $\kappa$ is called $\alpha$-Erdős if for every function $f:\kappa ^{<\omega }\to \{0,1\}$, there is a set of order type $\alpha$ that is homogeneous for $f$. In the notation of the partition calculus, $\kappa$ is $\alpha$-Erdős if

$\kappa \rightarrow (\alpha )^{<\omega }$.

The existence of zero sharp implies that the constructible universe $L$ satisfies "for every countable ordinal $\alpha$, there is an $\alpha$-Erdős cardinal". In fact, for every indiscernible $\kappa$, $L_{\kappa }$ satisfies "for every ordinal $\alpha$, there is an $\alpha$-Erdős cardinal in $\mathrm {Coll} (\omega ,\alpha )$" (the Lévy collapse to make $\alpha$ countable).

However, the existence of an $\omega _{1}$-Erdős cardinal implies existence of zero sharp. If $f$ is the satisfaction relation for $L$ (using ordinal parameters), then the existence of zero sharp is equivalent to there being an $\omega _{1}$-Erdős ordinal with respect to $f$. Thus, the existence of an $\omega _{1}$-Erdős cardinal implies that the axiom of constructibility is false.

The least $\omega$-Erdős cardinal is not weakly compact,^{p. 39.} nor is the least $\omega _{1}$-Erdős cardinal.^{p. 39}

If $\kappa$ is $\alpha$-Erdős, then it is $\alpha$-Erdős in every transitive model satisfying "$\alpha$ is countable."

See also

• List of large cardinal properties

References

• 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.