Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The diagonal reflection principle

Author: Sean Cox
Journal: Proc. Amer. Math. Soc. 140 (2012), 2893-2902
MSC (2010): Primary 03E05, 03E50, 03E57
Published electronically: June 2, 2011
MathSciNet review: 2910775
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a highly simultaneous version of stationary set reflection, called the Diagonal Reflection Principle (DRP). We prove that $ PFA^{+\omega_1}$ implies DRP, and DRP in turn implies that the nonstationary ideal on $ [\theta]^\omega$ condenses correctly for many structures. We also prove that MM implies a weaker version of DRP, which in turn implies that the nonstationary ideal on $ \theta \cap$   cof$ (\omega)$ condenses correctly for many structures.

References [Enhancements On Off] (What's this?)

  • [1] James E. Baumgartner, A new class of order types, Ann. Math. Logic 9 (1976), no. 3, 187-222. MR 0416925 (54:4988)
  • [2] James E. Baumgartner, Applications of the proper forcing axiom, Handbook of Set-Theoretic Topology North-Holland, Amsterdam, 1984, pp. 913-959. MR 776640 (86g:03084)
  • [3] Sean Cox, Covering theorems for the core model, and an application to stationary set reflection, Ann. Pure Appl. Logic 161 (2009), no. 1, 66-93. MR 2567927
  • [4] Qi Feng and Thomas Jech, Projective stationary sets and a strong reflection principle, J. London Math. Soc. (2) 58 (1998), no. 2, 271-283. MR 1668171 (2000b:03166)
  • [5] Matthew Foreman, Ideals and Generic Elementary Embeddings, Handbook of Set Theory, Springer, 2010.
  • [6] Matthew Foreman, Smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals, Adv. Math. 222 (2009), no. 2, 565-595. MR 2538021
  • [7] M. Foreman, M. Magidor, and S. Shelah, Martin's maximum, saturated ideals, and nonregular ultrafilters. I, Ann. of Math. (2) 127 (1988), no. 1, 1-47. MR 924672 (89f:03043)
  • [8] Leo Harrington and Saharon Shelah, Some exact equiconsistency results in set theory, Notre Dame J. Formal Logic 26 (1985), no. 2, 178-188. MR 783595 (86g:03079)
  • [9] John Krueger, On the weak reflection principle, to appear in Transactions of the American Mathematical Society.
  • [10] Paul Larson, Separating stationary reflection principles, J. Symbolic Logic 65 (2000), no. 1, 247-258. MR 1782117 (2001k:03094)
  • [11] Menachem Magidor, Reflecting stationary sets, J. Symbolic Logic 47 (1982), no. 4, 755-771 (1983). MR 683153 (84f:03046)
  • [12] W. Hugh Woodin, The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1999. MR 1713438 (2001e:03001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03E05, 03E50, 03E57

Retrieve articles in all journals with MSC (2010): 03E05, 03E50, 03E57

Additional Information

Sean Cox
Affiliation: Institut für Mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany

Received by editor(s): November 11, 2010
Received by editor(s) in revised form: March 2, 2011
Published electronically: June 2, 2011
Additional Notes: I thank Matt Foreman, Ralf Schindler, and Martin Zeman for helpful conversations on related topics.
Communicated by: Julia Knight
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society