Canonical structure in the universe of set theory: Part two

Annals of Pure and Applied Logic 142 (1):55-75 (2006)

Abstract

We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 211–243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness

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References found in this work

Squares, Scales and Stationary Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (01):35-98.
Scales, Squares and Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (1):35-98.
A Very Weak Square Principle.Matthew Foreman & Menachem Magidor - 1997 - Journal of Symbolic Logic 62 (1):175-196.

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Citations of this work

Aronszajn Trees and Failure of the Singular Cardinal Hypothesis.Itay Neeman - 2009 - Journal of Mathematical Logic 9 (1):139-157.
Diagonal Prikry Extensions.James Cummings & Matthew Foreman - 2010 - Journal of Symbolic Logic 75 (4):1383-1402.
On Singular Stationarity II.Omer Ben-Neria - 2019 - Journal of Symbolic Logic 84 (1):320-342.
The Tree Property at ℵ Ω+1.Dima Sinapova - 2012 - Journal of Symbolic Logic 77 (1):279-290.

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