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Measure-Theoretic Uniformity

In: Foundations of Mathematics

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  • Gerald E. Sacks

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.

Suggested Citation

  • Gerald E. Sacks, 1969. "Measure-Theoretic Uniformity," Springer Books, in: Jack J. Bulloff & Thomas C. Holyoke & Samuel W. Hahn (ed.), Foundations of Mathematics, pages 51-57, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-86745-3_6
    DOI: 10.1007/978-3-642-86745-3_6
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