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Martin’s maximum revisited

Martin’s maximum revisited We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM ++ makes the $${\Pi_2}$$ Π 2 -fragment of the theory of $${H_{\aleph_2}}$$ H ℵ 2 invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to $${H_{\aleph_2}}$$ H ℵ 2 of Woodin’s absoluteness results for $${L(\mathbb{R})}$$ L ( R ) . In due course of proving this, we shall give a new proof of some of these results of Woodin. Finally we relate our generic absoluteness results with the resurrection axioms introduced by Hamkins and Johnstone and with their unbounded versions introduced by Tsaprounis. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Martin’s maximum revisited

Archive for Mathematical Logic , Volume 55 (2) – Dec 11, 2015

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References (20)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-015-0466-3
Publisher site
See Article on Publisher Site

Abstract

We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM ++ makes the $${\Pi_2}$$ Π 2 -fragment of the theory of $${H_{\aleph_2}}$$ H ℵ 2 invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to $${H_{\aleph_2}}$$ H ℵ 2 of Woodin’s absoluteness results for $${L(\mathbb{R})}$$ L ( R ) . In due course of proving this, we shall give a new proof of some of these results of Woodin. Finally we relate our generic absoluteness results with the resurrection axioms introduced by Hamkins and Johnstone and with their unbounded versions introduced by Tsaprounis.

Journal

Archive for Mathematical LogicSpringer Journals

Published: Dec 11, 2015

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