The set omega, as the comment in this question points out, can be defined as the smallest set that is closed under succession and includes the empty set.
This is enough to define it uniquely, but to find it, that is to prove that it it exists, one must use the first axiom of infinity in ZFC.
Thus it appears that definition is not existence.
But, this doesn't seem quite right. For surely the axiom of infinity was introduced into ZFC such that an argument as shown above can have existential import.
In this line of thought, it appears that definition is enough for existence.
Is this right?
I would say not, and point to our coherent understanding and experience of the mathematical infinite that is condensed dually into definition and axiom.
Is this right as well?
(This, I think, appears to have certain tangential relationships with the various forms of the ontological arguments of God, which I don't want to go into here).
(It also appears to have some affinity with the notion of essence preceding existence).