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HIERARCHIES OF (VIRTUAL) RESURRECTION AXIOMS

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, USA E-mail:gunter.fuchs@csi.cuny.eduURL: www.math.csi.cuny.edu/∼fuchs
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Abstract

I analyze the hierarchies of the bounded resurrection axioms and their “virtual” versions, the virtual bounded resurrection axioms, for several classes of forcings (the emphasis being on the subcomplete forcings). I analyze these axioms in terms of implications and consistency strengths. For the virtual hierarchies, I provide level-by-level equiconsistencies with an appropriate hierarchy of virtual partially super-extendible cardinals. I show that the boldface resurrection axioms for subcomplete or countably closed forcing imply the failure of Todorčević’s square at the appropriate level. I also establish connections between these hierarchies and the hierarchies of bounded and weak bounded forcing axioms.

Keywords

03E0503E4003E5003E5503E57forcing axiomsresurrection axiomssubcomplete forcingsquare principlesremarkable cardinalsextendible cardinalsvirtually extendible cardinalsweak forcing axiomsbounded forcing axioms
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 283 - 325
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Apter, A. W., Removing Laver functions from supercompactness arguments. Mathematical Logic Quarterly, vol. 51 (2005), no. 2, pp. 154–156.CrossRefGoogle Scholar
Bagaria, J., Bounded forcing axioms as principles of generic absoluteness. Archive for Mathematical Logic, vol. 39 (2000), pp. 393–401.CrossRefGoogle 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
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 nonregular ultrafilters. Part I. Annals of Mathematics, vol. 127 (1988), no. 1, pp. 1–47.CrossRefGoogle Scholar
Fuchs, G., Closed maximality principles: Implications, separations and combinations, this Journal, vol. 73 (2008), no. 1, pp. 276–308.Google Scholar
Fuchs, G., Combined maximality principles up to large cardinals, this Journal, vol. 74 (2009), no. 3, pp. 1015–1046.Google Scholar
Fuchs, G., The subcompleteness of Magidor forcing. Archive for Mathematical Logic, 2017, to appear. Preprint available at http://www.math.csi.cuny.edu/∼fuchs/.Google Scholar
Fuchs, G., Hierarchies of forcing axioms, the continuum hypothesis and square principles, this Journal, vol. 83 (2018), no. 1, pp. 256–282.Google Scholar
Gitman, V., Ramsey-like cardinals, this Journal, vol. 76 (2011), no. 2, pp. 519–540.Google Scholar
Gitman, V. and Welch, P., Ramsey-like cardinals II, this Journal, vol. 76 (2011), no. 2, pp. 541–560.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., 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.CrossRefGoogle Scholar
Hamkins, J. D. and Johnstone, T. A., Strongly uplifting cardinals and the boldface resurrection axioms. Archive for Mathematical Logic, vol. 56 (2017), no. 7–8, pp. 1115–1133.CrossRefGoogle Scholar
Hayut, Y. and Lambie-Hanson, C., Simultaneous stationary reflection and square sequences. Preprint, 2016, arXiv:1603.05556.CrossRefGoogle Scholar
Jensen, R., The fine structure of the constructible hierarchy. Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.CrossRefGoogle 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: https://www.mathematik.hu-berlin.de/∼raesch/org/jensen.html.Google Scholar
Jensen, R. B., Subcomplete forcing and ${\rm {\cal L}}$-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, Singapore, 2014, pp. 83–182.CrossRefGoogle Scholar
Kunen, K., Set Theory. An Introduction to Independence Proofs, North Holland, Amsterdam, 1980.Google Scholar
Minden, K., On subcomplete forcing, Ph.D. thesis. The CUNY Graduate Center, 2017, preprint, arXiv:1705.00386 [math.LO].Google Scholar
Magidor, M. and Lambie-Hanson, C., On the strengths and weaknesses of weak squares, Appalachian Set Theory 2006–2012 (Cummings, J. and Schimmerling, E., editors), London Mathematical Society Lecture Notes Series, vol. 406, Cambridge University Press, Cambridge, 2013, pp. 301–330.Google Scholar
Moore, J. T., Set mapping reflection, this Journal, vol. 5 (2005), no. 1, pp. 87–97.Google Scholar
Schimmerling, E., Coherent sequences and threads. Advances in Mathematics, vol. 216 (2007), pp. 89–117.CrossRefGoogle Scholar
Stavi, J. and Väänänen, J., Reflection principles for the continuum, Logic and Algebra (Zhang, Y., editor), AMS Contemporary Mathematics Series, vol. 302, American Mathematical Society, Providence, RI, 2001.Google Scholar
Todorčević, S., Coherent sequences, Handbook of Set Theory, vol. 1, Springer, 2010, pp. 215–296.CrossRefGoogle Scholar
Tsaprounis, K., On resurrection axioms, this Journal, vol. 80 (2015), no. 2, pp. 587–608.Google Scholar
Viale, M., Audrito, G., and Steila, S., A boolean algebraic approach to semiproper iterations. Preprint, 2014, arXiv:1402.1714 [math.LO].Google Scholar
Veličković, B., Jensen’s $\square $ principles and the Novák number of partially ordered sets, this Journal, vol. 51 (1986), pp. 47–58.Google Scholar
Weiß, C., Subtle and ineffable tree properties, Ph.D. thesis, Ludwig-Maximilians-Universität München, 2010.Google Scholar