Closure properties of parametric subcompleteness

Archive for Mathematical Logic 57 (7-8):829-852 (2018)
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Abstract

For an ordinal \, I introduce a variant of the notion of subcompleteness of a forcing poset, which I call \-subcompleteness, and show that this class of forcings enjoys some closure properties that the original class of subcomplete forcings does not seem to have: factors of \-subcomplete forcings are \-subcomplete, and if \ and \ are forcing-equivalent notions, then \ is \-subcomplete iff \ is. I formulate a Two Step Theorem for \-subcompleteness and prove an RCS iteration theorem for \-subcompleteness which is slightly less restrictive than the original one, in that its formulation is more careful about the amount of collapsing necessary. Finally, I show that an adequate degree of \-subcompleteness follows from the \-distributivity of a forcing, for \.

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Citations of this work

Subcomplete forcing, trees, and generic absoluteness.Gunter Fuchs & Kaethe Minden - 2018 - Journal of Symbolic Logic 83 (3):1282-1305.
Aronszajn tree preservation and bounded forcing axioms.Gunter Fuchs - 2021 - Journal of Symbolic Logic 86 (1):293-315.
Subcomplete forcing principles and definable well‐orders.Gunter Fuchs - 2018 - Mathematical Logic Quarterly 64 (6):487-504.

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References found in this work

Saturated ideals.Kenneth Kunen - 1978 - Journal of Symbolic Logic 43 (1):65-76.
The subcompleteness of Magidor forcing.Gunter Fuchs - 2018 - Archive for Mathematical Logic 57 (3-4):273-284.

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