Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T19:15:38.571Z Has data issue: false hasContentIssue false
  • English
  • Français

Combined Maximality Principles up to large cardinals

Published online by Cambridge University Press:  12 March 2014

Gunter Fuchs
Gunter Fuchs*
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, Westfälische Wilhelms-Universität Münster, Einsteinstraβe 62, 48149 Münster, Germany, E-mail: gfuchs@uni-muenster.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the Maximality Principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 1015 - 1046
Copyright
Copyright © Association for Symbolic Logic 2009

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]Apter, Arthur W. and Hamkins, Joel David, Universal indestructibility, Kobe Journal of Mathematics, vol. 16 (1999), no. 2, pp. 119–130.Google Scholar
[2]Apter, Arthur W. and Hamkins, Joel David, Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata, Mathematical Logic Quarterly, vol. 47 (2001), no. 4, pp. 563–571.3.0.CO;2-#>CrossRefGoogle Scholar
[3]Foreman, Matthew, Handbook of set theory, Springer, chapter Ideals and Generic Elementary Embeddings.CrossRefGoogle Scholar
[4]Fuchs, Gunter, Closed maximality principles: Implications, separations and combinations, this Journal, vol. 73 (2008), no. 1, pp. 276–308.Google Scholar
[5]Fuchs, Gunter, Generic embeddings associated to an indestructibly weakly compact cardinal, submitted, ca. 30 pages.Google Scholar
[6]Hamkins, Joel David, Extensions with the approximation and cover properties have no new large cardinals, Fundamenta Mathematicae, vol. 180 (2003), no. 3, pp. 257–277.CrossRefGoogle Scholar
[7]Hamkins, Joel David, A simple maximality principle, this Journal, vol. 68 (2003), no. 2, pp. 527–550.Google Scholar
[8]Jech, Thomas, Set theory, the third millenium ed., Springer Monographs in Mathematics, Springer, Berlin, 2003.Google Scholar
[9]Jensen, Ronald B., Independence of the axiom of dependent choices from the countable axiom of choice, this Journal, vol. 31 (1966), p. 294.Google Scholar
[10]Jensen, Ronald B., Schimmerling, Ernest, Schindler, Ralf, and Steel, John R., Stacking mice, this Journal, vol. 74 (2009), pp. 315–335.Google Scholar
[11]Kanamori, Akihiro, The higher infinite, 2 ed., Springer Monographs in Mathematics, Springer, 2003.Google Scholar
[12]Leibman, George, Consistency strengths of modified maximality principles, Ph.D. thesis, The City University of New York, 2004.Google Scholar
[13]Menas, Telis K., On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1974), pp. 327–359.CrossRefGoogle Scholar
[14]Stavi, Jonathan and Väänänen, Jouko, Reflection principles for the continuum, Logic and Algebra, AMS Contemporary Mathematics Series, no. 302, 2001.Google Scholar
[15]Zeman, Martin, Inner models and large cardinals, de Gruyter, Berlin, New York, 2002.CrossRefGoogle Scholar