  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 phi
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 (diamond box phi) implies (box phi), 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 phi, is
equiconsistent with the scheme asserting that V_delta is an elementary
substructure of V for an inaccessible cardinal delta, which in turn is
equiconsistent with the scheme asserting that ORD is Mahlo. The strongest
principle along these lines is the Necessary Maximality Principle, which
asserts that the boldface 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.
