Absoluteness via resurrection

Journal of Mathematical Logic 17 (2):1750005 (2017)
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Abstract

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms for a class of forcings Γ and a given ordinal α), and show that RAω implies generic absoluteness for the first-order theory of Hγ+ with respect to forcings in Γ preserving the axiom, where γ = γΓ is a cardinal which depends on Γ. We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for Hγ0+ with respect to Γ0 and for Hγ1+ with respect to Γ1 with γ0 = γΓ0≠γΓ1 = γ1 is in principle possible, and we present several natural models o...

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References found in this work

The Higher Infinite.Akihiro Kanamori - 2000 - Studia Logica 65 (3):443-446.
The lottery preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
On strong compactness and supercompactness.Telis K. Menas - 1975 - Annals of Mathematical Logic 7 (4):327-359.
Forcing with Sequences of Models of Two Types.Itay Neeman - 2014 - Notre Dame Journal of Formal Logic 55 (2):265-298.
Suitable extender models II: Beyond ω-huge.W. Hugh Woodin - 2011 - Journal of Mathematical Logic 11 (2):115-436.

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