Measure-Theoretic Uniformity

Abstract

Here we present the principal ideas and results of [5] with some indications of proof. The general notion of uniformity is difficult to harvest; nonetheless, various offshoots of it have borne fruit in all fields of mathematical logic. In this paper we introduce the notion of measure-theoretic uniformity, and we describe its use in recursion theory, hyperarithmetic analysis, and set theory. In recursion theory we show that the set of all sets T such that the ordinals recursive in T are the recursive ordinals has measure 1. In set theory we obtain all of Cohen’s independence results [1,2]. Solovay [8,9] has extended Cohen’s method by forcing statements with closed, measurable sets of conditions rather than finite sets of conditions; in this manner he exploits the concepts of forcing and genericity to prove: if ZF is consistent, then ZF + “there exists a translation-invariant, countably additive extension of Lebesgue measure defined on all sets of reals” + “the dependent axiom of choice” is consistent. Solovay’s result is also a consequence of the notion of measure-theoretic uniformity.

The preparation of this paper was supported by U.S. Army Contract ARO-D-373. The author wishes to thank Professor Anil Nerode for many helpful conversations on uniformity and definability in set theory.

This paper is a slightly amplified version of a paper of the same name reprinted with permission of the publisher from the Bulletin of the American Mathematical Society; pp. 169–174; Copyright c. 1967, The American Mathematical Society.