Journal of Symbolic Logic

$\Pi^1_3$ Sets and $\Pi^1_3$ Singletons

Kai Hauser and W. Hugh Woodin

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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.

Article information

J. Symbolic Logic, Volume 64, Issue 2 (1999), 590-616.

First available in Project Euclid: 6 July 2007

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Zentralblatt MATH identifier


Primary: 03E15: Descriptive set theory [See also 28A05, 54H05]
Secondary: 04A15 03E60: Determinacy principles 03E45: Inner models, including constructibility, ordinal definability, and core models 03E55: Large cardinals 03E35: Consistency and independence results 03E60: Determinacy principles

Set Theory Descriptive Set Theory Core Models Definable Singletons


Hauser, Kai; Woodin, W. Hugh. $\Pi^1_3$ Sets and $\Pi^1_3$ Singletons. J. Symbolic Logic 64 (1999), no. 2, 590--616.

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