Diamonds, uniformization.

*(English)*Zbl 0598.03044This paper is concerned with the property \(\diamond_ S\) when GCH is assumed and when S is a stationary subset of \(\lambda^+\), where \(\lambda\) is a singular cardinal, or when \(S\subseteq \kappa\) and \(\kappa\) is strongly inaccessible. The work was begun in earlier papers by the author [see, e.g., Logic colloquium ’78, Proc., Mons/Belgium 1978, Stud. Logic Found. Math. 97, 357-380 (1979; Zbl 0449.03045)], and is substantially completed here. Thus if \(\lambda\) is singular and \(S\subseteq \{\delta <\lambda^+:\) cf \(\delta\) \(=cf \lambda \}\) then \(\diamond_ S\) may or may not hold. If \(\square_{\lambda}\) holds and F(S) is also stationary, where \(F(S)=\{\delta <\lambda^+:\) \(S\cap \delta\) is a stationary subset of \(\delta\) \(\}\), then \(\diamond_ S\) holds. This is proved using part of the proof of the strong covering lemma from Chapter XIII of the author’s book ”Proper forcing” (1982; Zbl 0495.03035). Further, for some S with \(F(S)=\emptyset\), \(\diamond_ S\) must still hold. But if F(S) is not stationary, a forcing construction (similar to one used by the author in his work on the failure of the Whitehead conjecture) can make \(\diamond_ S\) fail, and this in the strong form of a uniformization property.

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##### References:

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