Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-11T09:54:31.570Z Has data issue: false hasContentIssue false
  • English
  • Français

ON RESURRECTION AXIOMS

Published online by Cambridge University Press:  22 April 2015

KONSTANTINOS TSAPROUNIS
KONSTANTINOS TSAPROUNIS*
Affiliation:
DEPARTMENT OF LOGIC, HISTORY & PHILOSOPHY OF SCIENCE UNIVERSITY OF BARCELONA 08001 BARCELONA, SPAINE-mail: kostas.tsap@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The resurrection axioms are forms of forcing axioms that were introduced recently by Hamkins and Johnstone, who developed on earlier ideas of Chalons and Veličković. In this note, we introduce a stronger form of resurrection (which we call unbounded resurrection) and show that it gives rise to families of axioms which are consistent relative to extendible cardinals, and which imply the strongest known instances of forcing axioms, such as Martin’s Maximum++. In addition, we study the unbounded resurrection postulates in terms of consistency lower bounds, obtaining, for example, failures of the weak square principle.

Keywords

03E3503E5503E57Forcing axiomsResurrection axiomsExtendible cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 2 , June 2015 , pp. 587 - 608
Copyright
Copyright © The Association for Symbolic Logic 2015 

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

Bagaria, J., A characterization of Martin’s axiom in terms of absoluteness, this Journal, vol. 62 (1997), no. 2, pp. 366–372.Google Scholar
Bagaria, J., Bounded forcing axioms as principles of generic absoluteness. Archive for Mathematical Logic, vol. 39 (2000), no. 6, pp. 393–401.Google Scholar
Bagaria, J., C (n)-cardinals. Archive for Mathematical Logic, vol. 51 (2012), no. 3–4, pp. 213–240.Google Scholar
Bagaria, J., and Magidor, M., On ω 1-strongly compact cardinals. this Journal, vol. 79 (2014), no. 1, pp. 266–278.Google Scholar
Bagaria, J., Hamkins, J. D., Tsaprounis, K., and Usuba, T., Superstrong and other large cardinals are never Laver indestructible, preprint, 2013, http://arxiv.org/abs/1307.3486.Google Scholar
Cohen, P. J., The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America, vol. 50 (1963), pp. 1143–1148.Google Scholar
Corazza, P., Laver sequences for extendible and super-almost-huge cardinals, this Journal, vol. 64 (1999), no. 3, pp. 963–983.Google Scholar
Cox, S., PFA and ideals on ω2 whose associated forcings are proper. Notre Dame Journal of Formal Logic, vol. 53 (2012), no. 3, pp. 397–412.CrossRefGoogle Scholar
Cummings, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection. Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 35–98.Google Scholar
Eisworth, T., Successors of singular cardinals, Handbook of set theory (Foreman, M. and Kanamori, A., editors), Springer, 2010, pp. 1229–1350.CrossRefGoogle Scholar
Foreman, M., Ideals and generic elementary embeddings, Handbook of set theory (Foreman, M. and Kanamori, A., editors), Springer, 2010, pp. 885–1147.Google Scholar
Foreman, M., and Magidor, M., A very weak square principle, this Journal, vol. 62 (1997), no. 1, pp. 175–196.Google Scholar
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.CrossRefGoogle Scholar
Hamkins, J. D., A simple maximality principle, this Journal, vol. 68 (2003), no. 2, pp. 527–550.Google Scholar
Hamkins, J. D., and Johnstone, T. A., Resurrection axioms and uplifting cardinals. Archive for Mathematical Logic, vol. 53 (2014), no. 3–4, pp. 463–485.Google Scholar
Jech, T., Set theory, The Third Millennium Edition, Springer-Verlag, Berlin, 2002.Google Scholar
Kanamori, A., The higher infinite, Springer-Verlag, Berlin, 1994.Google Scholar
König, B., Forcing indestructibility of set-theoretic axioms, this Journal, vol. 72 (2007), no. 1, pp. 349–360.Google Scholar
Larson, P. B., The Stationary Tower, Notes on a Course by W. Hugh Woodin, University Lecture Series, vol. 32, American Mathematical Society, 2004.Google Scholar
Laver, R., Making the supercompactness of κ indestructible under κ-directed closed forcing. Israel Journal of Mathematics, vol. 29 (1978), pp. 385–388.CrossRefGoogle Scholar
Schindler, R., Semi-proper forcing, remarkable cardinals, and Bounded Martin’s maximum. Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 527–532.Google Scholar
Schindler, R., Bounded Martin’s maximum and strong cardinals, Set theory, Centre de recerca Matemàtica, Barcelona, 2003–2004 (Bagaria, J. and Todorčević, S., editors), Birkhäuser, 2006, pp. 401–406.CrossRefGoogle Scholar
Shelah, S., Proper and improper forcing, Springer-Verlag, Berlin, 1998.CrossRefGoogle Scholar
Solovay, R., and Tennenbaum, S., Iterated Cohen extensions and Souslin’s problem. Annals of Mathematics, vol. 94 (1971), pp. 201–245.Google Scholar
Stavi, J., and Väänänen, J., Reflection principles for the continuum, Logic and Algebra (Zhang, Yi, editor), Contemporary Mathematics, vol. 302, American Mathematical Society, 2002, pp. 59–84.Google Scholar
Tsaprounis, K., Elementary chains and C (n)-cardinals. Archive for Mathematical Logic, vol. 53 (2014), no. 1–2, pp. 89–118.Google Scholar
Tsaprounis, K., Large cardinals and resurrection axioms, Ph.D. dissertation, University of Barcelona, Spain, 2012.Google Scholar
Viale, M., Martin’s maximum revisited, preprint, 2013, www.personalweb.unito.it/matteo.viale/.Google Scholar
Viale, M., Category forcings, MM +++, and generic absoluteness for the theory of strong forcing axioms, preprint, 2013, www.personalweb.unito.it/matteo.viale/.Google Scholar
Woodin, W. H., The axiom of determinacy, forcing axioms, and the non-stationary ideal, 2nd ed., De Gruyter Series in Logic and its Applications, 2010.Google Scholar