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A simple maximality principle

  • Joel David Hamkins (a1) (a2)

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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[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
  • URL: /core/journals/journal-of-symbolic-logic
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