Hostname: page-component-cb9f654ff-lqqdg Total loading time: 0 Render date: 2025-08-24T06:35:04.165Z Has data issue: false hasContentIssue false
  • English
  • Français

A minimal counterexample to universal baireness

Published online by Cambridge University Press:  12 March 2014

Kai Hauser
Kai Hauser*
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA, E-mail: hauser@math.berkeley.edu Lehrstuhl Für Mathematische Logik, Humboldt Universität, 10099 Berlin, Germany, E-mail: Hauser@Mathematik.Hu-berlin.de
Get access Rights & Permissions [Opens in a new window]

Abstract

For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

Keywords

Set theorydescriptive set theoryinner modelsuniversally Baire sets

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 4 , December 1999 , pp. 1601 - 1627
Copyright
Copyright © Association for Symbolic Logic 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[Do] Dodd, A. J., The core model, London Mathematical Society Lecture Note Series, vol. 61, 1984, xxxviii+229.Google Scholar
[FeMaWo] Feng, Q., Magidor, M., and Woodin, H., Universally Baire sets of reals, Set theory of the continuum (Judah, H., Just, W., and Woodin, H., editors), no. 26, Mathematical Sciencies Research Institute Publications, 1992, pp. 203–242.CrossRefGoogle Scholar
[Ha1] Hauser, K., The consistency strength of projective absoluteness, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 245–295.CrossRefGoogle Scholar
[Ha2] Hauser, K., Sets of reals, infinite games and core model theory, Proceedings of the 6th Asian Logic Conference, Beijing 1996 (Chong, C. T.et al., editors), World Scientific : Singapore, 1998, pp. 107–120.Google Scholar
[HaHj] Hauser, K. and Hjorth, G., Strong cardinals in the core model, Annals of Pure and Applied Logic, vol. 83 (1997), pp. 165–198.CrossRefGoogle Scholar
[Je] Jech, T., Set theory, Academic Press, 1978, xi + 621.Google Scholar
[Ka] Kanamori, A., The higher infinite, Springer-Verlag, Berlin, 1995, xxiv + 536.Google Scholar
[KeMaSo] Kechris, A. S., Martin, D. A., and Solovay, R. M., Introduction to Q-theory, Cabal seminar 79-81, Lecture Notes in Mathematics, no. 1019, Springer-Verlag, 1983, pp. 199–282.CrossRefGoogle Scholar
[MaSo] Martin, D. A. and Solovay, R. M., A basis theorem for sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–160.CrossRefGoogle Scholar
[MaSt1] Martin, D. A. and Steel, J. R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.CrossRefGoogle Scholar
[MaSt2] Martin, D. A. and Steel, J. R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), pp. 1–73.CrossRefGoogle Scholar
[MiSt] Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, 1994, 130 pages.CrossRefGoogle Scholar
[Mo] Moschovakis, Y. N., Descriptive set theory, North-Holland, 1980, xii + 637.Google Scholar
[Sh] Shoenfield, J., The problem of predicativity, Essays on the foundations of mathematics (Hillel, Y. Baret al., editors), Magnes Press Jerusalem, 1961, pp. 132–142.Google Scholar
[So] Solovay, R. M., The independence of DC from AD, Cabal seminar 76-77, Lecture Notes in Mathematics, no. 689, Springer Verlag, 1978, pp. 171–183.CrossRefGoogle Scholar
[St1] Steel, J. R., The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer Verlag, 1996, 112 pages.CrossRefGoogle Scholar
[St2] Steel, J. R., Inner models with many Woodin cardinals, Annals of Pure and Applied Logic, vol. 65 (1993), pp. 185–209.CrossRefGoogle Scholar
[St3] Steel, J. R., Projectively wellordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77–104.CrossRefGoogle Scholar
[WoMaHa] Woodin, W. H., Mathias, A. R. D., and Hauser, K., The axiom of determinacy, monograph in preparation.Google Scholar