Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T05:38:31.103Z Has data issue: false hasContentIssue false
  • English
  • Français

The nonstationary ideal in the ℙmax extension

Published online by Cambridge University Press:  12 March 2014

Paul B. Larson
Paul B. Larson*
Affiliation:
Miami University, Department of Mathematics and Statistics, Oxford, Ohio 45056, USA. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The forcing construction ℙmax, invented by W. Hugh Woodin, produces a model whose collection of subsets of ω1 is in some sense maximal. In this paper we study the Boolean algebra induced by the nonstationary ideal on ω1 in this model. Among other things we show that the induced quotient does not have a simply definable form. We also prove several results about saturation properties of the ideal in this extension.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 1 , March 2007 , pp. 138 - 158
Copyright
Copyright © Association for Symbolic Logic 2007

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]Abraham, U., Proper forcing, Handbook of Set Theory (Foreman, M., Kanamori, A., and Magidor, M., editors), to appear.Google Scholar
[2]Asperó, D. and Welch, P., Bounded Martin's Maximum, weak Erdős cardinals and ψAC, this Journal, vol. 67 (2002), no. 3, pp. 1141–1152.Google Scholar
[3]Baumgartner, J. E., Hajnal, A., and Máté, A., Weak saturation properties of ideals, Infinite and Finite Sets, vol. I (Hajnal, A., Rado, R., and Sós, V. T., editors), North-Holland, Amsterdam, 1975, pp. 137–158.Google Scholar
[4]Foreman, M., Magidor, M., and Shelah, S., Martin's Maximum. saturated ideals, and non-regular ultrafilters. Part I, Annals of Mathematics, vol. 127 (1988), pp. 1–47.Google Scholar
[5]Goldring, N., Woodin cardinals and presaturated ideals. Annals of Pure and Applied Logic, vol. 55 (1992), no. 3, pp. 285–303.CrossRefGoogle Scholar
[6]Groszek, M. and Slaman, T.A., A basis theorem for perfect sets. The Bulletin of Symbolic Logic, vol. 4 (1998), no. 2, pp. 204–209.CrossRefGoogle Scholar
[7]Jech, T., Set Theory, the third millennium edition, revised and expanded ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[8]Jensen, R. M., Solovay, R. B., Some applications of almost disjoint sets, Mathematical Logic and Foundations of Set Theory (Proceedings of the International Colloquium, Jerusalem, 1968), North-Holland, Amsterdam, 1970, pp. 84–104.Google Scholar
[9]Kanamori, A., The higher infinite. Large cardinals in set theory from their beginnings, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994.Google Scholar
[10]Larson, P., A uniqueness theorem for iterations, this Journal, vol. 67 (2002), no. 4, pp. 1344–1350.Google Scholar
[11]Larson, P., The stationary tower. Notes on a course by W. Hugh Woodin, American Mathematical Society University Lecture Series, vol. 32, 2004.Google Scholar
[12]Larson, P., The canonical function game, Archive for Mathematical Logic, vol. 44 (2005), no. 7, pp. 581–595.CrossRefGoogle Scholar
[13]Larson, P., Forcing over models of determinacy, Handbook of Set Theory (Foreman, M., Kanamori, A., and Magidor, M., editors), to appear.Google Scholar
[14]Larson, P., Reals constructible from many countable sets of ordinals, preprint.Google Scholar
[15]Moore, J. T., A solution to the L-space problem and related ZFC constructions, preprint.Google Scholar
[16]Nyikos, P., Crowding offunctions, para-saturation of ideals, and topological applications, Topology Proceedings, vol. 28 (2004), no. 1, pp. 241–266.Google Scholar
[17]Shelah, S., Proper and improper forcing, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.CrossRefGoogle Scholar
[18]Steel, J. and Van Wesep, R., Two consequences of determinacy consistent with choice. Transactions of the American Mathematical Society, vol. 272 (1982), no. 1, pp. 67–85.CrossRefGoogle Scholar
[19]Taylor, A. D., Regularity properties of ideals and ultrafilters, Annals of Mathematical Logic, vol. 16 (1979), no. 1, pp. 33–55.CrossRefGoogle Scholar
[20]Todorcevic, S., Coherent sequences, Handbook of Set Theory (Foreman, M., Kanamori, A., and Magidor, M., editors), to appear.Google Scholar
[21]Todorcevic, S., A note on the proper forcing axiom, Axiomatic Set Theory (Boulder, Colorando), 1983, Also, in Contemporary Mathematics, vol. 31 (1984), American Mathematical Society, Providence, RI, pp. 209–218.Google Scholar
[22]Todorcevic, S., Partitioning pairs of countable ordinals, Acta Mathematica, vol. 159 (1987), no. 3–4, pp. 261–294.CrossRefGoogle Scholar
[23]Woodin, W.H., Some consistency results in ZFC using AD, Cabal Seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 172–198.Google Scholar
[24]Woodin, W.H., The axiom of determinacy, forcing axioms, and the nonstationary ideal, DeGruyter Series in Logic and its Applications, vol. 1, 1999.CrossRefGoogle Scholar
[25]Zapletal, J., Strongly almost disjoint functions, Israel Journal of Mathematics, vol. 97 (1997), pp. 101–111.CrossRefGoogle Scholar