Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T18:44:59.971Z Has data issue: false hasContentIssue false
  • English
  • Français

The perfect set theorem and definable wellorderings of the continuum

Published online by Cambridge University Press:  12 March 2014

Alexander S. Kechris
Alexander S. Kechris*
Affiliation:
California Institute of Technology, Pasadena, California 91125
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Γ be a collection of relations on the reals and let M be a set of reals. We call M a perfect set basis for Γ if every set in Γ with parameters from M which is not totally included in M contains a perfect subset with code in M. A simple elementary proof is given of the following result (assuming mild regularity conditions on Γ and M): If M is a perfect set basis for Γ, the field of every wellordering in Γ is contained in M. An immediate corollary is Mansfield's Theorem that the existence of a Σ21 wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 4 , December 1978 , pp. 630 - 634
Copyright
Copyright © Association for Symbolic Logic 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Fenstad, J. E., The axiom of determinateness, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J. E., Editor), North-Holland, Amsterdam, pp. 41–61.Google Scholar
[2]Kechris, A. S., On projective ordinals, this Journal, vol. 39 (1974), pp. 269–282.Google Scholar
[3]Kechris, A. S., The theory of countable analytical sets, Transactions of the American Mathematical Society, vol. 202(1975), pp. 259–297.CrossRefGoogle Scholar
[4]Kechris, A. S. and Moschovakis, Y. N., Two theorems about projective sets, Israel Journal of Mathematics, vol. 12(1972), pp. 391–399.CrossRefGoogle Scholar
[5]Mansfield, R., Perfect subsets of definable sets of real numbers, Pacific Journal of Mathematics, vol. 35(1970), pp. 451–457.CrossRefGoogle Scholar
[6]Mansfield, R., The nonexistence of 21 wellorderings of the Baire space, Fundamenta Mathematicae, vol. 86(1975), pp. 279–282.CrossRefGoogle Scholar
[7]Martin, D. A., Projective sets and cardinal numbers: some questions related to the continuum problem, this Journal (to appear).Google Scholar
[8]Moschovakis, Y. N., Descriptive set theory, North-Holland, Amsterdam (to appear).CrossRefGoogle Scholar
[9]Moschovakis, Y. N., Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory (Bar-Hillel, Y., Editor), North-Holland, Amsterdam, 1970, pp. 24–62.Google Scholar
[10]Moschovakis, Y. N., Uniformization in a playful universe, Bulletin of the American Mathematical Society, vol. 77(1970), pp. 731–736.CrossRefGoogle Scholar
[11]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[12]Shoenfield, J. R., Mathematical logic, Addison-Wesley, Reading, Massachusetts, 1967.Google Scholar
[13]Solovay, R. M., On the cardinality of 21 sets of reals, Foundations of mathematics, Symposium Commemorating the 60th Birthday of K. Gödel, Springer-Verlag, Berlin and New York, 1966, pp. 58–73.Google Scholar