Global co-stationarity of the ground model from a new countable length sequence
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Abstract:
Suppose $V\subseteq W$ are models of ZFC with the same ordinals, and that for all regular cardinals $\kappa$ in $W$, $V$ satisfies $\square _{\kappa }$. If $W\setminus V$ contains a sequence $r:\omega \rightarrow \gamma$ for some ordinal $\gamma$, then for all cardinals $\kappa <\lambda$ in $W$ with $\kappa$ regular in $W$ and $\lambda \ge \gamma$, $(\mathscr {P}_{\kappa }(\lambda ))^W\setminus V$ is stationary in $(\mathscr {P}_{\kappa }(\lambda ))^W$. That is, a new $\omega$-sequence achieves global co-stationarity of the ground model.References
- Uri Abraham and Saharon Shelah, Forcing closed unbounded sets, J. Symbolic Logic 48 (1983), no. 3, 643–657. MR 716625, DOI 10.2307/2273456
- Keith J. Devlin, Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1984. MR 750828, DOI 10.1007/978-3-662-21723-8
- Natasha Dobrinen and Sy-David Friedman, Internal consistency and global co-stationarity of the ground model, The Journal of Symbolic Logic (to appear).
- Natasha Dobrinen and Sy-David Friedman, Co-stationarity of the ground model, J. Symbolic Logic 71 (2006), no. 3, 1029–1043. MR 2251553, DOI 10.2178/jsl/1154698589
- Moti Gitik, Nonsplitting subset of ${\scr P}_\kappa (\kappa ^+)$, J. Symbolic Logic 50 (1985), no. 4, 881–894 (1986). MR 820120, DOI 10.2307/2273978
- Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
- David W. Kueker, Löwenheim-Skolem and interpolation theorems in infinitary languages, Bull. Amer. Math. Soc. 78 (1972), 211–215. MR 290942, DOI 10.1090/S0002-9904-1972-12921-5
- Menachem Magidor, Representing sets of ordinals as countable unions of sets in the core model, Trans. Amer. Math. Soc. 317 (1990), no. 1, 91–126. MR 939805, DOI 10.1090/S0002-9947-1990-0939805-5
- Telis K. Menas, On strong compactness and supercompactness, Ann. Math. Logic 7 (1974/75), 327–359. MR 357121, DOI 10.1016/0003-4843(75)90009-1
- Kanji Namba, Independence proof of $(\omega ,\,\omega _{\alpha })$-distributive law in complete Boolean algebras, Comment. Math. Univ. St. Paul. 19 (1971), 1–12. MR 297548
- Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
- Stevo Todorčević, Coherent sequences, Handbook of Set Theory (Matthew Foreman, Akihiro Kanamori, and Menachem Magidor, eds.), Kluwer (to appear).
Additional Information
- Natasha Dobrinen
- Affiliation: Kurt Gödel Research Center for Mathematical Logic, Währinger Strasse 25, 1090 Wien, Austria
- Address at time of publication: Department of Mathematics, University of Denver, Denver, Colorado 80208
- Email: dobrinen@logic.univie.ac.at. natasha.dobrinen@du.edu
- Received by editor(s): November 20, 2006
- Published electronically: January 9, 2008
- Additional Notes: This work was supported by FWF grant P 16334-N05. The author wishes to thank Justin Moore for invaluable help and Paul Larson for direction
- 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), 1815-1821
- MSC (2000): Primary 03E05, 03E35, 03E65, 05C05
- DOI: https://doi.org/10.1090/S0002-9939-08-09094-1
- MathSciNet review: 2373613