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*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 S 5. 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 δ ≺ 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 □ , 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.