Journal of Symbolic Logic

Universally Baire sets and definable well-orderings of the reals

Sy D. Friedman and Ralf Schindler

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Abstract

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

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

Dates
First available in Project Euclid: 31 October 2003

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1067620173

Digital Object Identifier
doi:10.2178/jsl/1067620173

Mathematical Reviews number (MathSciNet)
MR2017341

Zentralblatt MATH identifier
1055.03028

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

Keywords
Descriptive set theory large cardinals inner models

Citation

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. https://projecteuclid.org/euclid.jsl/1067620173


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References

  • R. David $\Delta^1_3$ reals, Annals of Mathematical Logic, vol. 23 (1982), pp. 121--125.
    Mathematical Reviews (MathSciNet): MR701123
    Digital Object Identifier: doi:10.1016/0003-4843(82)90002-X
    Zentralblatt MATH: 0519.03039
  • 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.
    Mathematical Reviews (MathSciNet): MR1233821
    Zentralblatt MATH: 0781.03034
  • S. D. Friedman David's trick, Proceedings of the 1997 asl summer meeting at leeds, vol. 258, Cambridge University Press,1999, pp. 67--71.
    Mathematical Reviews (MathSciNet): MR1720571
    Zentralblatt MATH: 0939.03052
  • K. Hauser The consistency strength of projective absoluteness, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 245--295.
    Mathematical Reviews (MathSciNet): MR1346111
    Digital Object Identifier: doi:10.1016/0168-0072(94)00041-Z
    Zentralblatt MATH: 0836.03025
  • K. Hauser and R.-D. Schindler Projective uniformization revisited, Annals of Pure and Applied Logic, vol. 103 (2000), pp. 109--153.
    Mathematical Reviews (MathSciNet): MR1756144
    Digital Object Identifier: doi:10.1016/S0168-0072(99)00038-X
    Zentralblatt MATH: 0961.03041
  • 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.
    Mathematical Reviews (MathSciNet): MR526913
    Zentralblatt MATH: 0397.03032
  • 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.
    Mathematical Reviews (MathSciNet): MR255391
    JSTOR: links.jstor.org
  • W. Mitchell and J. Steel Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, Berlin,1994.
    Mathematical Reviews (MathSciNet): MR1300637
    Zentralblatt MATH: 0805.03042
  • R. Schindler The core model for almost linear iterations, Annals of Pure and Applied Logic, vol. 116 (2002), pp. 207--274.
    Mathematical Reviews (MathSciNet): MR1900905
    Digital Object Identifier: doi:10.1016/S0168-0072(01)00113-0
    Zentralblatt MATH: 1017.03029
  • J. R. Steel Projectively well-ordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77--104.
    Mathematical Reviews (MathSciNet): MR1336414
    Digital Object Identifier: doi:10.1016/0168-0072(94)00021-T
    Zentralblatt MATH: 0821.03023
  • H. Woodin On the consistency strength of projective uniformization, Logic colloquium 81 (Amsterdam) (J. Stern, editor), North-Holland,1982, pp. 365--383.
    Mathematical Reviews (MathSciNet): MR757040
    Zentralblatt MATH: 0544.03024

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