A simple maximality principle

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 P and all subsequent extensions V P∗Q̇ 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 (3 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 Vδ ≺ 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 mp ∼ , which asserts that mp ∼ 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. 1 The Maximality Principle Christophe Chalons has introduced a delightful new axiom, asserting in plain language that anything that is forceable and not subsequently unforceable is true. Specifically, in [2] he proposes the following principle: My research has been supported by grants from the PSC-CUNY Research Foundation and the NSF. I would like to thank James Cummings, Ali Enayat, Ilijas Farah, Philip Welch and W. Hugh Woodin for helpful discussions concerning various topics arising in this article, as well as Paul Larson for introducing me to the topic and Christophe Chalons for his insightful email correspondence. Specifically, The College of Staten Island of CUNY and The CUNY Graduate Center. I am currently on leave as Visiting Associate Professor at Carnegie Mellon University, and I would like to thank the CMU Department of Mathematical Sciences for their hospitality.