  We present recent results on the model companions of set theory, placing them
in the context of the current debate in the philosophy of mathematics. We start
by describing the dependence of the notion of model companionship on the
signature, and then we analyze this dependence in the specific case of set
theory. We argue that the most natural model companions of set theory describe
(as the signature in which we axiomatize set theory varies) theories of
$H_{\kappa^+}$, as $\kappa$ ranges among the infinite cardinals. We also single
out $2^{\aleph_0}=\aleph_2$ as the unique solution of the Continuum problem
which can (and does) belong to some model companion of set theory (enriched
with large cardinal axioms). Finally this model-theoretic approach to
set-theoretic validities is explained and justified in terms of a form of
maximality inspired by Hilbert's axiom of completeness.

Logic