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

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

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.

Citations

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

[Rowbottom71-1] F. Rowbottom, "Some strong axioms of infinity incompatible with the axiom of constructibility". Annals of Mathematical Logic vol. 3, no. 1 (1971).

[1]

Fahrer / Team für die Suche eingeben:

Fahrer / Team für die Suche eingeben:

Langbahn Team – Weltmeisterschaft

Erdős cardinal

See also

References

Citations