Incompatible bounded category forcing axioms

Journal of Mathematical Logic 22 (2) (2022)
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Abstract

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We introduce bounded category forcing axioms for well-behaved classes [math]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [math] modulo forcing in [math], for some cardinal [math] naturally associated to [math]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [math] — to classes [math] with [math]. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [math]. We also show the existence of many classes [math] with [math] giving rise to pairwise incompatible theories for [math].

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David Aspero
University of East Anglia

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References found in this work

Set mapping reflection.Justin Tatch Moore - 2005 - Journal of Mathematical Logic 5 (1):87-97.
Resurrection axioms and uplifting cardinals.Joel David Hamkins & Thomas A. Johnstone - 2014 - Archive for Mathematical Logic 53 (3-4):463-485.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
Martin’s maximum revisited.Matteo Viale - 2016 - Archive for Mathematical Logic 55 (1-2):295-317.
Absoluteness via resurrection.Giorgio Audrito & Matteo Viale - 2017 - Journal of Mathematical Logic 17 (2):1750005.

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