Sets and singletons

Journal of Symbolic Logic 64 (2):590-616 (1999)

W. Hugh Woodin
Harvard University
We extend work of H. Friedman, L. Harrington and P. Welch to the third level of the projective hierarchy. Our main theorems say that (under appropriate background assumptions) the possibility to select definable elements of non-empty sets of reals at the third level of the projective hierarchy is equivalent to the disjunction of determinacy of games at the second level of the projective hierarchy and the existence of a core model (corresponding to this fragment of determinacy) which must then contain all real numbers. The proofs use Sacks forcing with perfect trees and core model techniques
Keywords Set Theory   Descriptive Set Theory   Core Models   Definable Singletons
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DOI 10.2307/2586486
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Some Descriptive Set Theory and Core Models.P. D. Welch - 1988 - Annals of Pure and Applied Logic 39 (3):273-290.
The Core Model.A. Dodd, R. Jensen, Tony Dodd, Ronald Jensen, A. J. Dodd & R. B. Jensen - 1984 - Journal of Symbolic Logic 49 (2):660-662.

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