On the ultrafilters and ultrapowers of strong partition cardinals

E-ISSN： 1943-5886|49|4|1268-1272

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.49, Iss.4, 1984-12, pp. : 1268-1272

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Abstract

A strong partition cardinal is an uncountable well-ordered cardinal κ such that every partition of [κ]^κ (the size κ subsets of κ) into less than κ many pieces has a homogeneous set of size κ. The existence of such cardinals is inconsistent with the axiom of choice, and our work concerning them is carried out in ZF set theory with just dependent choice (DC). The consistency of strong partition cardinals with this weaker theory remains an open question. The axiom of determinacy (AD) implies that a large number of cardinals including ℵ₁ have the strong partition property. The hypothesis that AD holds in the inner model of constructible sets built over the real numbers as urelements has important consequences for descriptive set theory, and results concerning strong partition cardinals are often applied in this context. Kechris [4] and Kechris et al. [5] contain further information concerning the relationship between AD and strong partition cardinals.