On the ultrafilters and ultrapowers of strong partition cardinals

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.We assume familiarity with the basic results on strong partition cardinals as developed in Kleinberg [6], [7], [8] and Henle [2]. Recall that a strong partition cardinal κ is measurable; in fact every stationary subset of κ is measure one under some normal measure on κ. If μ is a countably additive ultrafilter extending the closed unbounded filter on κ, then the length of the ultrapower [κ]κ under the less than almost everywhere μ ordering is again a measurable cardinal. In §1 below we establish a polarized partition property on these measurable cardinals.

Weak strong partition cardinals

In a series of papers [K2], [K3], [K4], E. M. Kleinberg established the extensive properties of what are now called “strong partition cardinals”, cardinals satisfying for all λ < κ. The purpose of this note is to show that all these consequences and the results in [H] and [W] can be obtained from the weaker relation and many from .We assume the reader is generally familiar with Kleinberg's machinery and with the definition of . We recall that a cardinal κ satisfies iff for every partition F: [κ]κ → A there is a p ∈ [κ]κ such that F″ [p]κ ≠ A. We take the liberty of regarding a p ∈ [κ]κ both as a subset of κ and as a function from κ to κ. We assume DC throughout.§1. From. Our results stem from the observation that the proofs in the papers cited above only require homogeneous sets for certain classes of partitions.Definition. A partition F: [κ]κ → λ, λ < λ, is called clopen if for all p ∈ [κ]κ there is an α < κ such that whenever -clopen is the assertion that all clopen partitions have homogeneous sets (Spector-Watro).

A measure representation theorem for strong partition cardinals

There are two main axiomatic extensions of Zermelo-Fraenkel set theory without the axiom of choice, that associated with the axiom of determinateness, and that associated with infinite exponent partition relations. Initially, the axiom of determinateness, henceforth AD, was the sole tool available. Using it, set theorists in the late 1960s produced many remarkable results in pure set theory (e.g. the measurability of ℵ1) as well as in projective set theory (e.g. reduction principles for ). Infinite exponent partition relations were first studied successfully soon after these early consequences of AD. They too produced measurable cardinals and not only were the constructions here easier than those from AD—the results gave a far clearer picture of the measures involved than had been offered by AD. In general, the techniques offered by infinite exponent partition relations became so attractive that a great deal of the subsequent work from AD involved an initial derivation from AD of the appropriate infinite exponent partition relation and then the derivation from the partition relation of the desired result.Since the early 1970s work on choiceless extensions of ZF + DC has split mainly between AD and its applications to projective set theory, and infinite exponent partition relations and their applications to pure set theory. There has certainly been a fair amount of interplay between the two, but for the most part the theories have been pursued independently.Unlike AD, infinite exponent partition relations have shown themselves amenable to nontrivial forcing arguments. For example, Spector has constructed models for interesting partition relations, consequences of AD, in which AD is false. Thus AD is a strictly stronger assumption than are various infinite exponent partition relations. Furthermore, Woodin has recently proved the consistency of infinite exponent partition relations relative to assumptions consistent with the axiom of choice, in particular, relative to the existence of a supercompact cardinal. The notion of doing this for AD is not even considered.