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*\Qdot 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
$(\possible\necessaryφ)\implies\necessaryφ$, 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 $Vδ\elesub V$
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
$\necessary\MPtilde$, which asserts that $\MPtilde$ 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.