Measure-theoretic uniformity in recursion theory and set theory
Gerald E. Sacks · Transactions of the American Mathematical Society · 1969
Introduction.The notion of measure-theoretic uniformity was presented in [23], [24].In this paper the notion is first utilized in an arithmetic setting and then extended to hyperarithmetic theory and set theory.The recursion-theoretic concepts touched on are : the arithmetic, hyperarithmetic and analytic hierarchies of Kleene [12]; co-models of the hyperarithmetic comprehension axiom of Kreisel [16]; recursive ordinals [11]; and hyperdegrees [13].One of the recursion-theoretic results we prove is: if P(X) is a fl{ predicate with free set variable X and the set XP(X) has positive Lebesgue measure, then P(A) holds for some hyperarithmetic set A. A corollary of this result is: the set of all A'such that the ordinals recursive in X coincide with the recursive ordinals has Lebesgue measure 1 (see footnote ( 9), below).In the area of set theory we are largely concerned with showing how relative consistency results follow in a natural manner from the notion of measure-theoretic uniformity.We develop Cohen's independence results [2] as well as a result of Solovay [29], [30] on the extendability of Lebesgue measure to all sets of reals.Solovay makes use of Cohen's forcing method, but he ingeniously replaces Cohen's finite forcing conditions by closed sets of positive measure.He shows: if ZF (Zermelo-Fraenkel set theory) is consistent, then ZF+ "there exists a translationinvariant, countably additive extension of Lebesque measure defined on all sets of reals"+ "the dependent axiom of choice" is consistent(2).We reprove Solovay's theorem with emphasis on the notion of measure-theoretic uniformity and with the help of some elementary properties of uniformly distributed, independent random variables; however, the fine details of our argument are not substantially different from those originated by Solovay.§4 of this paper, which deals solely with set theory, can be read independently of § §2 and 3, but it is intended to be read as a natural continuation of the earlier sections.