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Diamonds, uniformization. (English) Zbl 0598.03044
This 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.

##### MSC:
 3e+35 Consistency and independence results 3e+55 Large cardinals
##### Keywords:
diamond; GCH; stationary subset; forcing
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##### References:
  DOI: 10.1016/0003-4843(72)90001-0 · Zbl 0257.02035  Higher Souslin trees 41 pp 663– (1976)  Lecture Notes in Mathematics 405 (1974)  DOI: 10.1007/BF02762012 · Zbl 0403.03040  DOI: 10.1090/pspum/025/0357108  DOI: 10.1007/BF02762048 · Zbl 0451.03018  Logic Colloquium ’78 97 pp 357– (1979)  DOI: 10.1007/BF02759809 · Zbl 0369.02035  Proper forcing 940 (1982)  DOI: 10.1007/BF02760652 · Zbl 0467.03049
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