Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T18:21:23.067Z Has data issue: false hasContentIssue false
  • English
  • Français

HIERARCHIES OF FORCING AXIOMS, THE CONTINUUM HYPOTHESIS AND SQUARE PRINCIPLES

Published online by Cambridge University Press:  01 May 2018

GUNTER FUCHS
GUNTER FUCHS*
Affiliation:
THE COLLEGE OF STATEN ISLAND (CUNY) 2800 VICTORY BLVD. STATEN ISLAND, NY 10314, USA and THE GRADUATE CENTER (CUNY) 365 5TH AVENUE, NEW YORK NY10016, USAE-mail:gunter.fuchs@csi.cuny.eduURL: www.math.csi.cuny.edu/∼fuchs
Get access Rights & Permissions [Opens in a new window]

Abstract

I analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s ordinal reflection principle at ω2, and that its effect on the failure of weak squares is very similar to that of Martin’s Maximum.

Keywords

03E0503E4003E5003E5503E57forcing axiomssubcomplete forcingsquare principlesremarkable cardinalsweak forcing axiomsbounded forcing axioms
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 256 - 282
Copyright
Copyright © The Association for Symbolic Logic 2018 

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

Asperó, D., A maximal bounded forcing axiom, this Journal, vol. 67 (2002), no. 1, pp. 130–142.Google Scholar
Asperó, D. and Bagaria, J., Bounded forcing axioms and the continuum. Annals of Pure and Applied Logic, vol. 109 (2001), pp. 179–203.Google Scholar
Bagaria, J., Bounded forcing axioms as principles of generic absoluteness. Archive for Mathematical Logic, vol. 39 (2000), pp. 393–401.Google Scholar
Bagaria, J., Gitman, V., and Schindler, R., Remarkable cardinals, structural reflection, and the weak proper forcing axiom. Archive for Mathematical Logic, vol. 56 (2017), no. 1, pp. 1–20.CrossRefGoogle Scholar
Claverie, B. and Schindler, R., Woodin’s axiom (*), bounded forcing axioms, and precipitous ideals on ω 1, this Journal, vol. 77 (2012), no. 2, pp. 475–498.Google 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
Cummings, J. and Magidor, M., Martin’s maximum and weak square. Proceedings of the American Mathematical Society, vol. 139 (2011), no. 9, pp. 3339–3348.CrossRefGoogle 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), no. 1, pp. 1–47.Google Scholar
Fuchs, G., Generic embeddings associated to an indestructibly weakly compact cardinal. Annals of Pure and Applied Logic, vol. 162 (2010), no. 1, pp. 89–105.Google Scholar
Fuchs, G., The subcompleteness of Magidor forcing. Archive for Mathematical Logic, 2017. Preprint available at http://www.math.csi.cuny.edu/∼fuchs/SCM.pdf.Google Scholar
Goldstern, M. and Shelah, S., The bounded proper forcing axiom, this Journal, vol. 60 (1995), no. 1, pp. 58–73.Google Scholar
Hamkins, J. D., The lottery preparation. Annals of Pure and Applied Logic, vol. 101 (2000), no. 2–3, pp. 103–146.Google Scholar
Hayut, Y. and Karagila, A., Restrictions on forcings that change cofinalities. Archive for Mathematical Logic, vol. 55 (2016), no. 3, pp. 373–384.CrossRefGoogle Scholar
Hayut, Y. and Lambie-Hanson, C., Simultaneous stationary reflection and square sequences, arXiv preprint, 2016, arXiv:1603.05556v1.Google Scholar
Jech, T., Set Theory, Springer Monographs in Mathematics, Springer, Berlin, Heidelberg, 2003.Google Scholar
Jensen, R., The fine structure of the constructible hierarchy. Annals of Mathematical Logic, vol. 4, (1972), pp. 229–308.Google Scholar
Jensen, R. B., Forcing axioms compatible with CH, handwritten notes, 2009. Available at https://www.mathematik.hu-berlin.de/∼raesch/org/jensen.html.Google Scholar
Jensen, R. B., Subproper and subcomplete forcing, handwritten notes, 2009. Available at http://www.mathematik.hu-berlin.de/∼raesch/org/jensen.html.Google Scholar
Jensen, R. B., Subcomplete forcing and ${\cal L}$-forcing, E-Recursion, Forcing and C*-Algebras (Chong, C., Feng, Q., Slaman, T. A., Woodin, W. H., and Yang, Y., editors), Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 27, World Scientific, 2014, pp. 83–182.Google Scholar
Johnstone, T. A., Strongly unfoldable cardinals made indestructible, Ph.D thesis, The Graduate Center of the City University of New York, 2007.Google Scholar
Larson, P., Separating stationary reflection principles, this journal, vol. 65 (2000), no. 1, pp. 247–258.Google Scholar
Magidor, M., Combinatorial characterization of supercompact cardinals. Proceedings of the American Mathematical Society, vol. 42 (1974), no. 1, pp. 279–285.CrossRefGoogle Scholar
Magidor, M. and Lambie-Hanson, C., On the strengths and weaknesses of weak squares, Appalachian Set Theory 2006–2012, London Mathematical Society Lecture Notes Series, vol. 406, Cambridge University Press, 2013, pp. 301–330.Google Scholar
Minden, K., On subcomplete forcing, Ph.D thesis, The CUNY Graduate Center, 2017.Google Scholar
Miyamoto, T., A note on weak segments of PFA, Proceedings of the Sixth Asian Logic Conference (Chong, C., Feng, Q., Ding, D., Huang, Q., and Yasugi, M., editors), 1998, pp. 175–197.Google Scholar
Schimmerling, E., Coherent sequences and threads. Advances in Mathematics, vol. 216 (2007), pp. 89–117.Google Scholar
Schindler, R., Proper forcing and remarkable cardinals. Bulletin of Symbolic Logic, vol. 6 (2000), pp. 176–184.CrossRefGoogle Scholar
Schindler, R., Proper forcing and remarkable cardinals II, this Journal, vol. 66 (2001), pp. 1481–1492.Google Scholar
Todorčević, S., Trees and linearly ordered sets, Handbook of Set-Theoretic Topology, North Holland, 1984, pp. 235–293.Google Scholar
Tsaprounis, K., On resurrection axioms, this journal, vol. 80 (2015), no. 2, pp. 587–608.Google Scholar
Villaveces, A., Chains of end elementary extensions of models of set theory, this Journal, vol. 63 (1998), no. 3, pp. 1116–1136.Google Scholar