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  • Cited by 6
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Incurvati, Luca 2016. Maximality Principles in Set Theory. Philosophia Mathematica, p. nkw011.

    Hamkins, Joel David Leibman, George and Löwe, Benedikt 2015. Structural connections between a forcing class and its modal logic. Israel Journal of Mathematics, Vol. 207, Issue. 2, p. 617.

    Hamkins, Joel David and Johnstone, Thomas A. 2014. Resurrection axioms and uplifting cardinals. Archive for Mathematical Logic, Vol. 53, Issue. 3-4, p. 463.

    HAMKINS, JOEL DAVID 2012. THE SET-THEORETIC MULTIVERSE. The Review of Symbolic Logic, Vol. 5, Issue. 03, p. 416.

    Artemov, Sergei 2007. Handbook of Modal Logic.

    Hamkins, Joel D. and Hugh Woodin, W. 2005. The Necessary Maximality Principle for c. c. c. forcing is equiconsistent with a weakly compact cardinal. MLQ, Vol. 51, Issue. 5, p. 493.


A simple maximality principle

  • Joel David Hamkins (a1) (a2)
  • DOI:
  • Published online: 01 March 2014

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 φ holding in some forcing extension V and all subsequent extensions Vℙ*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 (◊ □ φ) ⇒ □ φ, 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 VδV 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 □ , which asserts that 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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[4]K. Hauser , The consistency strength of projective absoluteness. Annals of Pure and Applied Logic, vol. 74 (1995), no. 3, pp. 245295.

[6]M. Jorgensen . An equivalent form of LÉvy's Axiom Schema, Proceedings of the American Mathematical Society, vol. 26 (1970), no. 4, pp. 651654.

[7]B. Mitchell and E. Schimmerling , Covering without countable closure, Mathematical Research Letters, vol. 2 (1995), pp. 595609.

[10]J. Steel , The core model iterability problem. Lecture Notes in Logic, vol. 8, Springer-Verlag, 1996.

[13]W. H. Woodin , Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proceedings of the National Academy of Sciences, vol. 85 (1988), no. 18, pp. 65876591.

[14]W. H. Woodin , The Axiom of Determinacy, Forcing Axioms, and the Non-stationary Ideal, De Gruyter Series in Logic and its Applications, Walter de Gruyter, 1999.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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