Archive for Mathematical Logic

, Volume 55, Issue 1–2, pp 295–317 | Cite as

Martin’s maximum revisited

Article
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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 \({\Pi_2}\)-fragment of the theory of \({H_{\aleph_2}}\) invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to \({H_{\aleph_2}}\) of Woodin’s absoluteness results for \({L(\mathbb{R})}\). 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.

Keywords

Set theory Forcing axioms Generic absoluteness 

Mathematics Subject Classification

03E35 03E40 03E57 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Mathematical Logic, Dipartimento di MatematicaUniversità di TorinoTorinoItaly

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