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

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

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

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

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

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

Citations

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

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