 ZFC theorists wear their ontological commitments, either outright or conditional, very much on their postulational sleeve. They are out to characterize a very rich mathematical universe, to be sure—one in which there are so many things, and of such great variety of structure(s), that one should be able to find, therein, a set-theoretic ‘surrogate’ for virtually any kind of mathematical object or structure about which one may wish to make conjectures and prove theorems. With only this over-riding concern to unify all of mathematics within one overarching  domain,[31]  the ZFC theorist is not particularly committed to a logicist view of the objects and structures involved. If anything, logicism has acquired a new challenge: show how set theory itself—like arithmetic and analysis, say—is simply a body of logical truths in definitional disguise; and show how sets themselves can be (re-)construed as some kind of definitional concoction from purely logical  objects![32] 