Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T19:09:29.835Z Has data issue: false hasContentIssue false
  • English
  • Français

Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω1

Published online by Cambridge University Press:  24 February 2017

Benjamin Claverie  and
Ralf Schindler
Benjamin Claverie
Affiliation:
Institut Für Mathematische Logik Und Grundlagenforschung, Universität MünsterEinsteinstr. 62, 48149 Münster, Germany, E-mail: claverie@skl8em.comE-mail:, rds@math.uni-muenster.de
Ralf Schindler
Affiliation:
Institut Für Mathematische Logik Und Grundlagenforschung, Universität MünsterEinsteinstr. 62, 48149 Münster, Germany, E-mail: claverie@skl8em.comE-mail:, rds@math.uni-muenster.de
Get access Rights & Permissions [Opens in a new window]

Abstract

If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at ℵ2 with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC-model. This yields that if Woodin's ℙmax axiom (*) holds, then BPFA implies that V is closed under the “Woodin-in-the-next-ZFC-model” operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus “NSω1 is precipitous” and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor, [2, Question 6.11].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 475 - 498
Copyright
Copyright © Association for Symbolic Logic 2012

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.)

References

REFERENCES

[1] Bagaria, J., Bounded forcing axioms as principles of generic absoluteness, Archive for Mathematical Logic, vol. 39 (2000), pp. 393–401.CrossRefGoogle Scholar
[2] Baumgartner, J.E. and Taylor, A.D., Saturation properties of ideals in forcing extensions. II, Transactions of the American Mathematical Society, vol. 271 (1982), pp. 587–609.Google Scholar
[3] Busche, D. and Schindler, R., The strength of choiceless patterns of singular and weakly compact cardinals, Annals of Pure and Applied Logic, vol. 159 (2009), pp. 198–248.CrossRefGoogle Scholar
[4] Claverie, B. and Schindler, R., Increasing u2 by a stationary set preserving forcing, this Journal, vol. 74 (2009), pp. 187–200.Google Scholar
[5] Doebler, Ph. and Schindler, R., Π2 consequences of BMM plus NS is precipitous and the semiproperness of all stationary set preserving forcings, Mathematical Research Letters, vol. 16 (2009), no. 5, pp. 797–815.CrossRefGoogle Scholar
[6] Feng, C., Magidor, M., and Woodin, W.H., Universally Baire sets of reals, Set theory of the continuum (Judah, et al., editor), MSRI Publications, vol. 26, Springer-Verlag, Heidelberg, 1992, pp. 203–242.CrossRefGoogle Scholar
[7] Fuchs, G., Neeman, I., and Schindler, R., A criterion for coarse iterability. Archive for Mathematical Logic, vol. 49 (2010), pp. 447–468.CrossRefGoogle Scholar
[8] Goldstern, M. and Shelah, S., The hounded proper forcing axiom, this Journal, vol. 60 (1995), pp. 58–73.Google Scholar
[9] Hjorth, G., The influence of u2 , Ph.D. thesis, UC Berkeley, 1993.Google Scholar
[10] Jech, T., Set theory. Springer-Verlag, 2002.Google Scholar
[11] Jensen, R., The fine structure of the constructible hierarchy. With an appendix of J. Silver, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.CrossRefGoogle Scholar
[12] Jensen, R., Schimmerling, E., Schindler, R., and Steel, J., Stacking mice, this Journal, vol. 74 (2009), pp. 315–335.Google Scholar
[13] Jensen, R. and Steel, J., In Preparation.Google Scholar
[14] Ketchersid, R., Larson, P., and Zapletal, J., Increasing and Namba style forcing, this Journal. vol. 72 (2007), pp. 1372–1378.Google Scholar
[15] Mitchell, W. and Schimmerling, E., Weak covering without countable closure, Mathematical Research Letters, vol. 2 (1995), pp. 595–609.CrossRefGoogle Scholar
[16] Schimmerling, E., Combinatorial principles in the core model for one Woodin cardinal. Annals of Pure and Applied Logic, vol. 74 (1995), pp. 153–201.CrossRefGoogle Scholar
[17] Schimmerling, E. and Steel, J., The maximality of the core model. Transactions of the American Mathematical Society, vol. 351 (1999), pp. 3119–3141.CrossRefGoogle Scholar
[18] Schimmerling, E. and Zeman, M.. Square in core models. The Bulletin of Symbolic Logic, vol. 7 (2001), pp. 305–314.CrossRefGoogle Scholar
[19] Schimmerling, E., Characterization of ֊ i in core models, Journal of Mathematical Logic, vol. 4 (2004), pp. 1–72.CrossRefGoogle Scholar
[20] Schindler, R., Proper forcing and remarkable cardinals II, this Journal, vol. 66 (2001), pp. 1481–1492.Google Scholar
[21] Schindler, R., Semi-proper forcing, remarkable cardinals, and bounded Martin s maximum. Mathematical Logic Quarterly, vol. 50 (2004), pp. 5270–532.CrossRefGoogle Scholar
[22] Schindler, R., Iterates of the core model, this Journal, vol. 71 (2006), pp. 241–251.Google Scholar
[23] Schindler, R. and Steel, J., The self-iterability of L[E], this Journal, vol. 74 (2009), pp. 751–779.Google Scholar
[24] Shelah, S., Iterated forcing and normal ideals on ω1 , Israel journal of Mathematics, vol. 60 (1987), pp. 345–380.CrossRefGoogle Scholar
[25] Steel, J., The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer-Verlag, 1993.Google Scholar
[26] Steel, J., Projectively well-ordered inner models. Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77–104.CrossRefGoogle Scholar
[27] Steel, J., Outline of inner model theory, Handbook of set theory (Foreman, and Kanamori, , editors). Springer-Verlag, 2010, pp. 1595–1684.CrossRefGoogle Scholar
[28] Steel, J. and Zoble, S., Determinacy from strong reflection, submitted.Google Scholar
[29] Todorčević, S., A note on the proper forcing axiom. Axiomatic set theory (Boulder, Colorado, 1983), Contemporary Mathematics, vol. 31, American Mathematical Society, 1983, pp. 209–218.CrossRefGoogle Scholar
[30] Todorčević, S., Localized reflection and fragments of PFA, Logic and scientific methods, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 259, American Mathematical Society, 1997, pp. 145–155.CrossRefGoogle Scholar
[31] Woodin, W.H., The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter, 1999.CrossRefGoogle Scholar
[32] Zeman, M., Inner models and large cardinals, de Gruyter Series in Logic and its Application, vol. 5, de Gruyter, Berlin, New York, 2002.CrossRefGoogle Scholar