A simple maximality principle

Journal of Symbolic Logic 68 (2):527-550 (2003)

Joel David Hamkins
Oxford University
In this paper, following an idea of Christophe Chalons. I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence varphi holding in some forcing extension $V^P$ and all subsequent extensions $V^{P\ast Q}$ holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\lozenge \square \varphi) \Rightarrow \square \varphi$ , and is equivalent to the modal theory S5. In this article. I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that $V_\delta \prec V$ for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is $\square MP\!_{\!\!\!\!\!\!_{\!\!_\sim}}$ , which asserts that $MP\!_{\!\!\!\!\!\!_{\!\!_\sim}}$ holds in V and all forcing extensions. From this, it follows that $0^\#$ exists, that $x^\#$ exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain
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DOI 10.2178/jsl/1052669062
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The Consistency Strength of Projective Absoluteness.Kai Hauser - 1995 - Annals of Pure and Applied Logic 74 (3):245-295.

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The Set-Theoretic Multiverse.Joel David Hamkins - 2012 - Review of Symbolic Logic 5 (3):416-449.
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Internal Consistency and the Inner Model Hypothesis.Sy-David Friedman - 2006 - Bulletin of Symbolic Logic 12 (4):591-600.

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