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sets and singletons

Published online by Cambridge University Press:  12 March 2014

Kai Hauser  and
W. Hugh Woodin
Kai Hauser
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720., USA E-mail: hauser@math.berkeley.edu
W. Hugh Woodin
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720., USA E-mail: woodin@math.berkeley.edu
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Abstract

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 theorydescriptive set theorycore modelsdefinable singletons
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999 , pp. 590 - 616
Copyright
Copyright © Association for Symbolic Logic 1999

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