On a fragment of the universal Baire property for $\Sigma ^1_2$ sets
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Abstract:
There is a well-known global equivalence between $\Sigma ^1_2$ sets having the universal Baire property, two-step $\Sigma ^1_3$ generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of $\Sigma ^1_2$ sets being $(2^{\omega })^{+}$-cc-universally Baire, which is below $0^{\#}$. In a model obtained, there is a $\Sigma ^1_2$ set which is weakly $\omega _2$-universally Baire but not $\omega _2$-universally Baire.References
- Qi Feng, Menachem Magidor, and Hugh Woodin, Universally Baire sets of reals, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 203–242. MR 1233821, DOI 10.1007/978-1-4613-9754-0_{1}5
- Leo Harrington and Saharon Shelah, Some exact equiconsistency results in set theory, Notre Dame J. Formal Logic 26 (1985), no. 2, 178–188. MR 783595, DOI 10.1305/ndjfl/1093870823
- Thoralf Räsch and Ralf Schindler, A new condensation principle, Arch. Math. Logic 44 (2005), no. 2, 159–166. MR 2121257, DOI 10.1007/s00153-004-0227-1
- Ralf Schindler, Forcing axioms and projective sets of reals, Classical and new paradigms of computation and their complexity hierarchies, Trends Log. Stud. Log. Libr., vol. 23, Kluwer Acad. Publ., Dordrecht, 2004, pp. 207–222. MR 2156740, DOI 10.1007/978-1-4020-2776-5_{1}2
- Ralf-Dieter Schindler, Proper forcing and remarkable cardinals, Bull. Symbolic Logic 6 (2000), no. 2, 176–184. MR 1765054, DOI 10.2307/421205
- Ralf-Dieter Schindler, Proper forcing and remarkable cardinals. II, J. Symbolic Logic 66 (2001), no. 3, 1481–1492. MR 1856755, DOI 10.2307/2695120
- Todorcevic, S., Zoble, S., Baire reflection, Trans. Amer. Math. Soc. (to appear).
- Woodin, H., On the strength of projective uniformization, Logic Colloquium ’81, J. Stern (ed.), 1982, pp. 365-383.
- Stuart Zoble, Stationary reflection and the universal Baire property, Fund. Math. 191 (2006), no. 1, 45–56. MR 2232195, DOI 10.4064/fm191-1-3
Additional Information
- Stuart Zoble
- Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
- Email: szoble@math.toronto.edu
- Received by editor(s): March 20, 2006
- Received by editor(s) in revised form: September 12, 2006
- Published electronically: January 17, 2008
- Communicated by: Julia Knight
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 1807-1814
- MSC (2000): Primary 03E45; Secondary 03E35
- DOI: https://doi.org/10.1090/S0002-9939-08-08918-1
- MathSciNet review: 2373612