Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T14:34:42.438Z Has data issue: false hasContentIssue false
  • English
  • Français

MRP, tree properties and square principles

Published online by Cambridge University Press:  12 March 2014

Remi Strullu
Remi Strullu*
Affiliation:
Equipe de Logique Mathématique, Université Paris Diderot Paris 7,UFR de Mathématiques Case 7012, Site Chevaleret, 75205 Paris Cedex 13, France, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that MRP + MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2. This generalizes a result by Weiβ who showed that PFA implies that ITP(λ, ω2) holds for all cardinal λ ≥ ω2. Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω1) for all λ ≥ ω2.

Keywords

MRPITPsquare sequencestrongly compact cardinalconsistency results
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1441 - 1452
Copyright
Copyright © Association for Symbolic Logic 2011

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]Caicedo, Andrés Eduardo and Veličković, Boban, The bounded proper forcing axiom and well orderings of the reals, Mathematical Research Letters, vol. 13 (2006), no. 2–3, pp. 393–408.CrossRefGoogle Scholar
[2]Devlin, Keith J., The Yorkshireman's guide to proper forcing. Surveys in set theory (Mathias, A. R. D., editor), London Mathematical Society Lecture Note Series, vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 60–115.CrossRefGoogle Scholar
[3]Foreman, M., Magidor, M., and Shelah, S., Martin's maximum, saturated ideals, and nonregular ultrafilters. I, Annals of Mathematics. Second Series, vol. 127 (1988), no. 1, pp. 1–47.CrossRefGoogle Scholar
[4]Jech, Thomas, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[5]Jensen, R. Björn, The fine structure of the constructive hierarchy, Annals of Pure and Applied Logic, vol. 4 (1972), pp. 229–308; erratum, ibid. 4 (1972), 443, with a section by Jack Silver.Google Scholar
[6]König, Bernhard, Forcing indestructibility of set-theoretic axioms, this Journal, vol. 72 (2007), no. 1, pp. 349–360.Google Scholar
[7]Kunen, Kenneth, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1980.Google Scholar
[8]Moore, Justin Tatch, Set mapping reflection, Journal of Mathematical Logic, vol. 5 (2005), no. 1, pp. 87–97.CrossRefGoogle Scholar
[9]Moore, Justin Tatch, A five element basis for the uncountable linear orders. Annals of Mathematics. Second Series, vol. 163 (2006), no. 2, pp. 669–688.CrossRefGoogle Scholar
[10]Schimmerling, Ernest, Combinatorial principles in the core model for one Woodin cardinal, Annals of Pure and Applied Logic, vol. 74 (1995), no. 2, pp. 153–201.CrossRefGoogle Scholar
[11]Viale, Matteo, The proper forcing axiom and the singular cardinal hypothesis, this Journal, vol. 71 (2006), no. 2, pp. 473–479.Google Scholar
[12]Viale, Matteo and Weiβ, Christoph, On the consistency strength of the proper forcing axiom. Advances in Mathematics, to appear.Google Scholar
[13]Weiβ, Christoph, Subtle and ineffable tree properties, Ph.D. thesis, Ludwig Maximilians Universität München, 2010, http://edoc.ub.uni-muenchen.de/11438/.Google Scholar
[14]Weiβ, Christoph, The combinatorial essence of supercompactness, forthcoming.Google Scholar