##
**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.

Full Text:
DOI

### References:

[1] | DOI: 10.1016/0003-4843(72)90001-0 · Zbl 0257.02035 |

[2] | Higher Souslin trees 41 pp 663– (1976) |

[3] | Lecture Notes in Mathematics 405 (1974) |

[4] | DOI: 10.1007/BF02762012 · Zbl 0403.03040 |

[5] | DOI: 10.1090/pspum/025/0357108 |

[6] | DOI: 10.1007/BF02762048 · Zbl 0451.03018 |

[7] | Logic Colloquium ’78 97 pp 357– (1979) |

[8] | DOI: 10.1007/BF02759809 · Zbl 0369.02035 |

[9] | Proper forcing 940 (1982) |

[10] | DOI: 10.1007/BF02760652 · Zbl 0467.03049 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.