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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
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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

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References

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