Abstract
We present S. Shelah’s result thatS 21 ={δ<ω 2: cf(δ)=ω 1} may have the uniformization property (cf., §1, or [3] for a definition) for “well-chosen sequences”, 〈η δ:δ∈S 21 ^η δ an increasingω 1-sequence of ordinals converging to δ〉. This implies that\(GCH\not \to \diamondsuit _{s_1^2 } \), which shows that Gregory’ result (cf., [2]),\(GCH \to \diamondsuit _{s_0^2 } \), is the best possible.
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References
K. Devlin and S. Shelah,A weak version of ♦ which follows from \(2^{\kappa _0 }< 2^{\kappa _1 } \), Israel J. Math.29 (1978), 239–247.
J. Gregory,Higher Souslin trees and the Generalized Continuum Hypothesis. J. Symbolic Logic41 (1976), 663–671.
S. Shelah,Whitehead groups may not be free, even assuming CH, I, Israel J. Math.28 (1977), 193–204.
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Steinhorn, C.I., King, J.H. The uniformization property for ℵ2 . Israel J. Math. 36, 248–256 (1980). https://doi.org/10.1007/BF02762048
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DOI: https://doi.org/10.1007/BF02762048