Definability of measures and ultrafilters

Abstract

Nonprincipal ultrafilters are harder to define in ZFC, and harder to obtain in ZF + DC, than nonprincipal measures.The function μ from P(X) to the closed interval [0, 1] is a measure on X if μ. is finitely additive on disjoint sets and μ(X) = 1. (P is the power set.) μ is nonprincipal if μ ({x}) = 0 for each x Є X. μ is an ultrafilter if Range μ= {0, 1}. The existence of nonprincipal measures and ultrafilters on any infinite X follows from the axiom of choice.Nonprincipal measures cannot necessarily be defined in ZFC. (ZF is Zermelo–Fraenkel set theory. ZFC is ZF with choice.) In ZF alone they cannot even be proved to exist. This was first established by Solovay [14] using an inaccessible cardinal. In the model of [14] no nonprincipal measure on ω is even ODR (definable from ordinal and real parameters). The HODR (hereditarily ODR) sets of this model form a model of ZF + DC (dependent choice) in which no nonprincipal measure on ω exists. Pincus [8] gave a model with the same properties making no use of an inaccessible. (This model was also known to Solovay.) The second model can be combined with ideas of A. Blass [1] to give a model of ZF + DC in which no nonprincipal measures exist on any set. Using this model one obtains a model of ZFC in which no nonprincipal measure on the set of real numbers is ODR. H. Friedman, in private communication, previously obtained such a model of ZFC by a different method. Our construction will be sketched in 4.1.