Originally Posted by

**MoeBlee** What does it mean to say "there NEED exist only"? To be clear about all this, I would suggest formulating such questions in rigorous mathematical terms. It's too vague, at least for me, to talk about what "needs" to be.

First, what does it mean to say "a certain set is implied by ZF" or even "the existence of a certain set is implied by ZF". ZF proves existence theorems. Those are sentences of the form ExP, and given a model of ZF, there is a member of the universe of the model such that P is true, per that model, of that member.