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Universally Baire sets and definable well-orderings of the reals

Published online by Cambridge University Press:  12 March 2014

SY D. Friedman  and
Ralf Schindler
SY D. Friedman
Affiliation:
Institut Für Formale Logik, Universität Wien, Währinger STR. 25, 1090 Wien, Austria, E-mail: [email protected]
Ralf Schindler
Affiliation:
Institut Für Formale Logik, Universität Wien, Währinger STR. 25, 1090 Wien, Austria, E-mail: [email protected]
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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.

Keywords

Descriptive set theorylarge cardinalsinner models
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1065 - 1081
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] David, R., Δ3 1 reals. Annals of Mathematical Logic, vol. 23 (1982), pp. 121–125.CrossRefGoogle Scholar
[2] David, R., A very absolute Π2 1-singleton, Annals of Mathematical Logic, vol. 23 (1982), pp. 101–120.CrossRefGoogle Scholar
[3] Feng, Q., Magidor, M., and Woodin, H., 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.CrossRefGoogle Scholar
[4] Friedman, S. D., David's trick, Proceedings of the 1997 asl summer meeting at leeds, vol. 258, Cambridge University Press, 1999, pp. 67–71.Google Scholar
[5[ Friedman, S. D., Fine structure and class forcing, de Gruyter Series in Logic and its Applications, vol. 3, Walter de Gruyter & Co., Berlin, New York, 2000.CrossRefGoogle Scholar
[6] Friedman, S. D., Genericity and large cardinals, to appear.Google Scholar
[7] Hauser, K., The consistency strength of projective absoluteness, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 245–295.CrossRefGoogle Scholar
[8] Hauser, K. and Schindler, R.-D., Projective uniformization revisited, Annals of Pure and Applied Logic, vol. 103 (2000), pp. 109–153.CrossRefGoogle Scholar
[9] Jech, T., Set theory, San Diego, 1978.Google Scholar
[10] Kechris, A. S. and Moschovakis, Y. N., Notes on the theory of scales, Cabal seminar 76–77 (Kechris, A. S. and Moschovakis, Y. N., editors), Lecture Notes in Math., vol. 689, Berlin, 1978, pp. 1–53.CrossRefGoogle Scholar
[11] Martin, D. A. and Solovay, R. M., A basis theorem for Σ3 1 sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–160.CrossRefGoogle Scholar
[12] Mitchell, W. and Steel, J., Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, Berlin, 1994.CrossRefGoogle Scholar
[13] Schindler, R., The core model for almost linear iterations, Annals of Pure and Applied Logic, vol. 116 (2002), pp. 207–274.CrossRefGoogle Scholar
[14] Steel, J. R., Projectively well-ordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77–104.CrossRefGoogle Scholar
[15] Steel, J. R., The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer-Verlag, Berlin, 1996.CrossRefGoogle Scholar
[16] Woodin, H., On the consistency strength of projective uniformization, Logic colloquium 81 (Amsterdam) (Stern, J., editor), North-Holland, 1982, pp. 365–383.Google Scholar