Measure-theoretic Uniformity

Here we present the principal ideas and results of [5] with some indications of proof. 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 [l] , [2] without any use, overt or concealed, of his method of forcing or his notion of genericity. 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 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 countable axiom of choice" is consistent. Solovay's result is also a consequence of the notion of measure-theoretic uniformity. We begin with the simplest possible example of measure-theoretic uniformity. Let T be an arbitrary set of natural numbers, and let P be the power set of the natural numbers. We think of P as the product of countably many copies of a two-point set {a, b}. We assign the unbiased measure: tn({a, &}) = 1, m({a}) =m({b}) = J, and m((j>) = 0. We give P the induced product measure denoted by u. Let R(T, x, y) be a recursive predicate of the set-variable T and the number variables x and y. A familiar uniformity can be expressed as follows: If for some given T we have (x)(Ey)R(T, x, y)y then there exists a function ƒ recursive in the given T such that (x)R(T, x,f(x)). Before we introduce the measure-theoretic counterpart of this uniformity, we must shift our point of view from Skölem functions to bounding functions in order to make the measure come out right: if for some given T we have (x)(Ey)R(T, x, y), then there exists a function/recursive in T such that (x)(Ey)yûf(X)R(Tf x, y). Note that the existence of a Skölem function is equivalent to the existence of a bounding function. It is not hard to verify: if { T\ (x)(Ey)R(T> x, y) \ has measure 1, then { T\ (£ƒ)(ƒ recursive and (x)(Ey)yèf(X)R(Ti x, y)) ) has measure 1. Thus the restriction of the bounding function ƒ to the