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Not collapsing cardinals ≤κ in (<κ)-support iterations

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Abstract

We deal with the problem of preserving various versions of completeness in (<κ)-support iterations of forcing notions, generalizing the case “S-complete proper is preserved by CS iterations for a stationary costationaryS⊆ω 1”. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinalκ and a stationary setS⊆κ with fat complement we can have uniformization for every (A δ :δ ∈S′),A δ ⊆δ = supA δ , cf (δ) = otp(A δ ) and a stationary non-reflecting setS′⊆S (see B.8.2).

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, 91904, Jerusalem, Israel

    Saharon Shelah

  2. Department of Mathematics, Rutgers University, 08854, New Brunswick, NJ, USA

    Saharon Shelah

  3. Mathematics Department, University of Wisconsin - Madison, 53706, Madison, WI, USA

    Saharon Shelah

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  1. Saharon Shelah
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Correspondence to Saharon Shelah.

Additional information

Research supported by The German-Israeli Foundation for Scientific Research & Development Grant No. G-294.081.06/93 and by The National Science Foundation Grant No. 144-EF67. Publication No. 587.

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Shelah, S. Not collapsing cardinals ≤κ in (<κ)-support iterations. Isr. J. Math. 136, 29–115 (2003). https://doi.org/10.1007/BF02807192

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  • DOI: https://doi.org/10.1007/BF02807192

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