  We introduce and study some variants of a notion of canonical set theoretical
truth. By this, we mean truth in a transitive proper class model $M$ of ZFC
that is uniquely characterized by some $\in$-formula. We show that there are
interesting statements that hold in all such models, but do not follow from
ZFC, such as the ground model axiom and the nonexistence of measurable
cardinals. We also study a related concept in which we only require M to be
fixed up to elementary equivalence. We show that this theory-canonicity also
goes beyond provability in ZFC, but it does not rule out measurable cardinals
and it does not fix the size of the continuum.
