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The Perfect Set Theorem and Definable Wellorderings of the Continuum

Kechris, Alexander S. (1978) The Perfect Set Theorem and Definable Wellorderings of the Continuum. Journal of Symbolic Logic, 43 (4). pp. 630-634. ISSN 0022-4812.

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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 Σ^1_2 wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given.

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Additional Information:© 1979, Association for Symbolic Logic. Received November 15, 1976. Research partially supported by NSF Grant MPS75-07562.
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MathSciNet ReviewMR0518668
Zentralblatt MATH Identifier0401.03023
Issue or Number:4
Record Number:CaltechAUTHORS:20130528-081912845
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Official Citation:The Perfect Set Theorem and Definable Wellorderings of the Continuum Alexander S. Kechris; 630-634
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38673
Deposited By: Ruth Sustaita
Deposited On:28 May 2013 15:34
Last Modified:03 Oct 2019 04:59

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