Measure-Theoretic Uniformity

  • 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.

Keywords

Anil 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cohen, P. J.: The independence of the continuum hypothesis I. Proc. nat. Acad. Sci. 50, 1143–1148 (1963).MathSciNetCrossRefGoogle Scholar
  2. 2.
    — The independence of the continuum hypothesis II. Proc. nat. Acad. Sci. 51, 105–110 (1964).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Feferman, S.: Some applications of the notions of forcing and generic sets. Fund. Math. 56, 325–345 (1965).MathSciNetMATHGoogle Scholar
  4. 4.
    Kreisel, G.: The axiom of choice and the class of hyperarithmetic functions. Indag. Math. 24, 307–319 (1962).MathSciNetGoogle Scholar
  5. 5.
    Sacks, G. E.: Measure-theoretic uniformity in recursion theory and set theory. Trans. Amer. Math. Soc, to appear.Google Scholar
  6. 6.
    — On the fundamental equivalence type of a countable model. In preparation.Google Scholar
  7. 7.
    Scott, D., and R. Solovay: Boolean-valued models and forcing. To appear.Google Scholar
  8. 8.
    Solovay, R.: The measure problem, Abstract 65T-62. Not. Amer. Math. Soc. 12, 217 (1965).Google Scholar
  9. 9.
    — The measure problem. To appear.Google Scholar
  10. 10.
    Spector, C.: Measure-theoretic construction of incomparable hyperdegrees. J. Symb. Log. 23, 280–288 (1958).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Tanaka, H.: Some results in the effective descriptive set theory. Publ. RIMS, Kyoto Univ. Ser A, 3 II-52 (1967).Google Scholar
  12. 12.
    Tanaka, H. — A basis result for II 11 sets of positive measure. Mimeographed notes, Hosei University 1967.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1969

Authors and Affiliations

  • Gerald E. Sacks

There are no affiliations available

Personalised recommendations