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Erdős cardinal

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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 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

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References

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  • 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

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