More on the Revised GCH and the Black Box

Annals of Pure and Applied Logic 140 (1):133-160 (2006)
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We strengthen the revised GCH theorem by showing, e.g., that for , for all but finitely many regular κ ω implies that the diamond holds on λ when restricted to cofinality κ for all but finitely many .We strengthen previous results on the black box and the middle diamond: previously it was established that these principles hold on for sufficiently large n; here we succeed in replacing a sufficiently large n with a sufficiently large n.The main theorem, concerning the accessibility of λ on cofinality κ, Theorem 3.1, implies as a special case that for every regular λ>ω, for some κ<ω, we can find a sequence such that , , and we can fix a finite set of “exceptional” regular cardinals θ<ω so that if Aλ satisfies A<ω, there is a pair-coloring so that for every -monochromatic BA with no last element, letting δ:=supB it holds that —provided that is not one of the finitely many “exceptional” members of



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Citations of this work

Bounds for Covering Numbers.Andreas Liu - 2006 - Journal of Symbolic Logic 71 (4):1303 - 1310.
Zf + dc + ax4.Saharon Shelah - 2016 - Archive for Mathematical Logic 55 (1-2):239-294.
Higher Miller forcing may collapse cardinals.Heike Mildenberger & Saharon Shelah - 2021 - Journal of Symbolic Logic 86 (4):1721-1744.

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References found in this work

Reflecting stationary sets and successors of singular cardinals.Saharon Shelah - 1991 - Archive for Mathematical Logic 31 (1):25-53.
Applications of PCF theory.Saharon Shelah - 2000 - Journal of Symbolic Logic 65 (4):1624-1674.
Diamonds, uniformization.Saharon Shelah - 1984 - Journal of Symbolic Logic 49 (4):1022-1033.
Set theory without choice: not everything on cofinality is possible.Saharon Shelah - 1997 - Archive for Mathematical Logic 36 (2):81-125.

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