Bounded stationary reflection

Cummings, James; Lambie-Hanson, Chris

doi:10.1090/proc12743

Bounded stationary reflection
HTML articles powered by AMS MathViewer

by James Cummings and Chris Lambie-Hanson

Proc. Amer. Math. Soc. 144 (2016), 861-873

DOI: https://doi.org/10.1090/proc12743

Published electronically: June 26, 2015

PDF | Request permission

Abstract:

We prove that, assuming large cardinals, it is consistent that there are many singular cardinals $\mu$ such that every stationary subset of $\mu ^+$ reflects but there are stationary subsets of $\mu ^+$ that do not reflect at ordinals of arbitrarily high cofinality. This answers a question raised by Todd Eisworth.

References

Y. Chayut, Stationary reflection and the approachability property, Master’s thesis, Hebrew University of Jerusalem, 2013.
Todd Eisworth, Private communication.
Todd Eisworth, Successors of singular cardinals, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 1229–1350. MR 2768694, DOI 10.1007/978-1-4020-5764-9_{1}6
Todd Eisworth, Simultaneous reflection and impossible ideals, J. Symbolic Logic 77 (2012), no. 4, 1325–1338. MR 3051629, DOI 10.2178/jsl.7704160
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR 309729, DOI 10.1016/0003-4843(72)90001-0
Menachem Magidor, Reflecting stationary sets, J. Symbolic Logic 47 (1982), no. 4, 755–771 (1983). MR 683153, DOI 10.2307/2273097
Saharon Shelah, Reflecting stationary sets and successors of singular cardinals, Arch. Math. Logic 31 (1991), no. 1, 25–53. MR 1126352, DOI 10.1007/BF01370693
Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), no. 1, 73–116. MR 482431, DOI 10.1016/0003-4843(78)90031-1