Advertisement

Archive for Mathematical Logic

, Volume 30, Issue 3, pp 171–180 | Cite as

Measure theory and weak König's lemma

  • Xiaokang Yu
  • Stephen G. Simpson
Article

Abstract

We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak König's lemma) and prove that it is strictly weaker thanWKL (weak König's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space is countably additive on open sets.

Keywords

Mathematical Logic Lebesgue Measure Formal Version Measure Theory Borel Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brown, D.K., Simpson, S.G.: Which set existence axioms are needed to prove the separable Hahn-Banach theorem? Ann. Pure Appl. Logic31, 123–144 (1986)Google Scholar
  2. 2.
    Halmos, P.R.: Measure theory. New York: Van Nostrand 1950Google Scholar
  3. 3.
    Jockusch, C.G., Jr.:Π Π 10 classes and Boolean combinations of recursively enumerable sets. J. Symb. Logic39, 95–96 (1974)Google Scholar
  4. 4.
    Jockusch, C.G., Soare, R.I.:Π 10 classes and degrees of theories. Trans. Am. Math. Soc.173, 33–56 (1972)Google Scholar
  5. 5.
    Rudin, W.: Real and complex analysis. New York: McGraw-Hill 1966Google Scholar
  6. 6.
    Sacks, G.E.: Degrees of Unsolvability. Ann. Math. Stud.55, Princeton University Press, 1963Google Scholar
  7. 7.
    Simpson, S.G.: Subsystems of Second Order Arithmetic. (in preparation)Google Scholar
  8. 8.
    Simpson, S.G.: Subsystems of Z2 and reverse mathematics, appendix to Proof theory, second edition, by G. Takeuti. Amsterdam: North-Holland, 1986, pp. 434–448Google Scholar
  9. 9.
    Simpson, S.G.: Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations? J. Symb. Logic49, 783–802 (1984)Google Scholar
  10. 10.
    Yu, X.: Measure theory in weak subsystems of second order arithmetic. Ph.D. Thesis, The Pennsylvania State University, 1987Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Xiaokang Yu
    • 1
  • Stephen G. Simpson
    • 2
  1. 1.Department of MathematicsPennsylvania State UniversityAltoonaUSA
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

Personalised recommendations