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Generic Σ31 absoluteness

Published online by Cambridge University Press:  12 March 2014

Sy D. Friedman
Affiliation:
Institute for Logic, University of Vienna, Waehringer Strasse 25, A-1090 Vienna, Austria Mathematics Department, MIT, Cambridge. Massachusetts 02139, USA, E-mail: sdf@logic.univie.ac.at
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Extract

In this article we study the strength of absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [3]. (In particular, see Theorem 3 below.) We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA.

The statement “ absoluteness holds for ccc forcing” means that if a formula with real parameters has a solution in a ccc set-forcing extension of the universe V, then it already has a solution in V. The analogous definition applies when ccc is replaced by other set-forcing notions, or by class-forcing.

Theorem 1. [1] absoluteness for ccc has no strength; i.e., if ZFC is consistent then so is ZFC + absoluteness for ccc.

The following results concerning (arbitrary) set-forcing and class-forcing can be found in [3].

Theorem 2 (Feng-Magidor-Woodin). (a) absoluteness for arbitrary set-forcing is equiconsistent with the existence of a reflecting cardinal, i.e., a regular cardinal κ such that H(κ) is2-elementary in V.

(b) absoluteness for class-forcing is inconsistent.

We consider next the following set-forcing notions, which lie strictly between ccc and arbitrary set-forcing: proper, semiproper, stationary-preserving and ω1-preserving. We refer the reader to [8] for the definitions of these forcing notions.

Using a variant of an argument due to Goldstern-Shelah (see [6]), we show the following. This result corrects Theorem 2 of [3] (whose proof only shows that if absoluteness holds in a certain proper forcing extension, then in L either ω1 is Mahlo or ω2 is inaccessible).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2004

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References

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