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A relative of the approachability ideal, diamond and non-saturation

  • Assaf Rinot (a1)

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that together with 2λ = λ+ implies ⋄S for every S ⊆ λ+ that reflects stationarily often.

In this paper, for a set S ⊆ λ+, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved:

1. if I[S; λ] is fat, then NSλ + ∣ S is non-saturated;

2. if I[S; λ] is fat and 2λ = λ+, then ⋄S holds;

3. implies that I[S; λ] is fat for every Sλ+ that reflects stationarily often;

4. it is relatively consistent with the existence of a supercompact cardinal that fails, while I[S; λ] is fat for every stationary S ⊆ λ+ that reflects stationarily often.

The stronger principle is studied as well.

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[1] James Cummings , Matthew Foreman , and Menachem Magidor , Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 3598.

[3] Keith J. Devlin and Hávard Johnsbráten , The Souslin problem, Lecture Notes in Mathematics, Vol. 405, Springer-Verlag, Berlin, 1974.

[4] Mirna Džamonja and Saharon Shelah , Saturatedfilters at successors of singular, weak reflection and yet another weak club principle. Annals of Pure and Applied Logic, vol. 79 (1996), no. 3, pp. 289316.

[7] Moti Gitik and Saharon Shelah , Less saturated ideals, Proceedings of the American Mathematical Society, vol. 125 (1997), no. 5, pp. 15231530.

[9] Leo Harrington and Saharon Shelah , Some exact equiconsistency results in set theory, Notre Dame Journal of Formal Logic, vol. 26 (1985), no. 2, pp. 178188.

[16] Justin Tatch Moore , The proper forcing axiom, Prikry forcing, and the singular cardinals hypothesis, Annals of Pure and Applied Logic, vol. 140 (2006), no. 1-3, pp. 128132.

[17] Assaf Rinot , A cofinality-preserving smallforcing may introduce a special aronszajn tree, Archive for Mathematical Logic, vol. 48 (2009), no. 8, pp. 817823.

[22] Saharon Shelah , Diamonds, Proceedings of the American Mathematical Society, vol. 138 (2010), pp. 21512161.

[23] Charles I. Steinhorn and James H. King , The uniformization property for ℵ2, Israeljournal of Mathematics, vol. 36 (1980), no. 3–4, pp. 248256.

[25] Martin Zeman , Diamond, GCH and weak square, Proceedings of the American Mathematical Society, vol. 138 (2010), no. 5, pp. 18531859.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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