Erdős cardinal explained

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .

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 → (\alpha) ^{<\omega} .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 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.^{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. . James E. Baumgartner . Galvin . Fred . Fred Galvin . Generalized Erdős cardinals and 0^# . 10.1016/0003-4843(78)90012-8 . 528659 . 1978 . Annals of Mathematical Logic . 0003-4843 . 15 . 3 . 289–313 . free .
• Book: Drake, F. R.. Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. 1974. 0-444-10535-2.
• Erdős . Paul . Hajnal . András . On the structure of set-mappings . 10.1007/BF02023868 . free . 0095124 . 1958 . . 0001-5954 . 9 . 1–2 . 111–131 . 18976050 .
• Book: Kanamori, Akihiro. 2003. Springer. The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings. The Higher Infinite . Akihiro Kanamori. 2nd. 3-540-00384-3.

Citations

Notes and References

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