Reflecting stationary sets and successors of singular cardinals

Archive for Mathematical Logic 31 (1):25-53 (1991)

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for manyλ ⌝ PT(λ, ℵ1). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular λ which is a limit of supercompacts the bad stationary set concentrates on the “right” cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular λ {δ<+ : cfδ<λ} is good. It is proved in ZFC that ifλ=cfλ>ℵ1 then {α<+ : cfα<λ} is the union of λ sets on which there are squares
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DOI 10.1007/BF01370693
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References found in this work BETA

Full Reflection of Stationary Sets Below ℵω.Thomas Jech & Saharon Shelah - 1990 - Journal of Symbolic Logic 55 (2):822 - 830.
The Primal Framework II: Smoothness.J. T. Baldwin & S. Shelah - 1991 - Annals of Pure and Applied Logic 55 (1):1-34.
Independence Results.Saharon Shelah - 1980 - Journal of Symbolic Logic 45 (3):563-573.
On Power of Singular Cardinals.Saharon Shelah - 1986 - Notre Dame Journal of Formal Logic 27 (2):263-299.

View all 6 references / Add more references

Citations of this work BETA

Squares, Scales and Stationary Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (01):35-98.
A Model of Cummings and Foreman Revisited.Spencer Unger - 2014 - Annals of Pure and Applied Logic 165 (12):1813-1831.
Adding Closed Unbounded Subsets of Ω₂ with Finite Forcing.William J. Mitchell - 2005 - Notre Dame Journal of Formal Logic 46 (3):357-371.
Applications of PCF Theory.Saharon Shelah - 2000 - Journal of Symbolic Logic 65 (4):1624-1674.

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