## Journal of Symbolic Logic

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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745795

Mathematical Reviews number (MathSciNet)
MR1777772

Zentralblatt MATH identifier
0930.03057

JSTOR
Hauser, Kai; Woodin, W. Hugh. $\Pi^1_3$ Sets and $\Pi^1_3$ Singletons. J. Symbolic Logic 64 (1999), no. 2, 590--616. https://projecteuclid.org/euclid.jsl/1183745795