Journal of Symbolic Logic

Universally Baire sets and definable well-orderings of the reals

Sy D. Friedman and Ralf Schindler

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Let n ≥ 3 be an integer. We show that it is consistent (relative to the consistency of n-2 strong cardinals) that every σ1n-set of reals is universally Baire yet there is a (lightface) projective well-ordering of the reals. The proof uses “David’s trick” in the presence of inner models with strong cardinals.

Article information

J. Symbolic Logic, Volume 68, Issue 4 (2003), 1065-1081.

First available in Project Euclid: 31 October 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E35: Consistency and independence results 03E45: Inner models, including constructibility, ordinal definability, and core models 03E55: Large cardinals

Descriptive set theory large cardinals inner models


Friedman, Sy D.; Schindler, Ralf. Universally Baire sets and definable well-orderings of the reals. J. Symbolic Logic 68 (2003), no. 4, 1065--1081. doi:10.2178/jsl/1067620173.

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  • R. David $\Delta^1_3$ reals, Annals of Mathematical Logic, vol. 23 (1982), pp. 121--125.
  • Q. Feng, M. Magidor, and H. Woodin Universally Baire sets of reals, Set theory of the continuum (Judan et al., editors), Mathematical Sciences Research Institute Publications, vol. 26, Springer Verlag,1992, pp. 203--242.
  • S. D. Friedman David's trick, Proceedings of the 1997 asl summer meeting at leeds, vol. 258, Cambridge University Press,1999, pp. 67--71.
  • K. Hauser The consistency strength of projective absoluteness, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 245--295.
  • K. Hauser and R.-D. Schindler Projective uniformization revisited, Annals of Pure and Applied Logic, vol. 103 (2000), pp. 109--153.
  • T. Jech Set theory, San Diego,1978.
  • A. S. Kechris and Y. N. Moschovakis Notes on the theory of scales, Cabal seminar 76--77 (A. S. Kechris and Y. N. Moschovakis, editors), Lecture Notes in Math., vol. 689, Berlin,1978, pp. 1--53.
  • D. A. Martin and R. M. Solovay A basis theorem for $\Sigma^1_3$ sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138--160.
  • W. Mitchell and J. Steel Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, Berlin,1994.
  • R. Schindler The core model for almost linear iterations, Annals of Pure and Applied Logic, vol. 116 (2002), pp. 207--274.
  • J. R. Steel Projectively well-ordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77--104.
  • H. Woodin On the consistency strength of projective uniformization, Logic colloquium 81 (Amsterdam) (J. Stern, editor), North-Holland,1982, pp. 365--383.