Abstract
It is well-known that the square principle \({\square_\lambda}\) entails the existence of a non-reflecting stationary subset of λ+, whereas the weak square principle \({\square^{*} _\lambda}\) does not. Here we show that if μcf(λ) < λ for all μ < λ, then \({\square^{*} _\lambda}\) entails the existence of a non-reflecting stationary subset of \({E^{\lambda^+}_{{\rm cf}(\lambda)}}\) in the forcing extension for adding a single Cohen subset of λ+.
It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of \({\square^{*} _\lambda}\) for every singular cardinal λ of countable cofinality.
Similar content being viewed by others
References
Cummings J., Foreman M., Magidor M.: Squares, scales and stationary reflection. J. Math. Log. 1, 35–98 (2001)
Cummings J., Magidor M.: Martin’s maximum and weak square. Proc. Amer. Math. Soc. 139, 3339–3348 (2011)
Devlin K.J.: Variations on \({\diamondsuit}\), J. Symbolic Logic. 44, 51–58 (1979)
G. Fuchs, Hierarchies of forcing axioms, the continuum hypothesis and square principles, J. Symbolic Logic (to appear); preprint available at www.math.csi.cuny.edu/~fuchs/.
G. Fuchs, Closure properties of parametric subcompleteness, submitted in 2017; preprint available at www.math.csi.cuny.edu/~fuchs/ .
R. B. Jensen, Subcomplete forcing and \({\mathcal{L}}\) -forcing, in: E-Recursion, Forcing and C*-Algebras, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 27, World Sci. Publ. (Hackensack, NJ, 2014), pp. 83–182.
R. B. Jensen, Forcing axioms compatible with CH, Handwritten notes, available at www.mathematik.hu-berlin.de/~raesch/org/jensen.html (2009).
R. B. Jensen, Subproper and subcomplete forcing, Handwritten notes, available at www.mathematik.hu-berlin.de/~raesch/org/jensen.html (2009).
K. Kunen, Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co. (Amsterdam, 1983). An introduction to independence proofs, reprint of the 1980 original.
Larson P.: Separating stationary reflection principles. J. Symbolic Logic. 65, 247–258 (2000)
A. Rinot, A relative of the approachability ideal, diamond and non-saturation, J. Symbolic Logic, 75 (2010) 1035–1065.
J. Silver, The independence of Kurepa’s conjecture and two-cardinal conjectures in model theory, in: Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967), Amer. Math. Soc. (Providence, RI, 1971), pp. 383–390.
S. Todorčević, Trees and linearly ordered sets, in: Handbook of Set-theoretic Topology, North-Holland (Amsterdam, 1984), pp. 235–293.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was partially supported by PSC-CUNY grant 69656-00 47.
The second author was partially supported by the Israel Science Foundation (grant # 1630/14).
Rights and permissions
About this article
Cite this article
Fuchs, G., Rinot, A. Weak square and stationary reflection. Acta Math. Hungar. 155, 393–405 (2018). https://doi.org/10.1007/s10474-018-0789-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-018-0789-8