TITLE: When we say, "ZFC can found most of mathematics," what do we really mean?
QUESTION [15 upvotes]: ZFC works as a foundation because it can prove many sentences that are "translations" of theorems from "standard" mathematics into the language of ZFC.
But there's a subtlety. When we say, "ZFC can found most of mathematics," what do we really mean?
Do we mean

ZFC proves most theorems (suitably translated into the language of sets) in the mathematics literature
A metatheory that makes sense of "consistency" + the assumption that ZFC is consistent can be used to prove most theorems (suitably translated into the language of sets) in the mathematics literature
A metatheory that makes sense of "models" + the assumption that ZFC has a model can be used to prove most theorems (suitably translated into the language of sets) in the mathematics literature
A metatheory that makes sense of "models" + the assumption that ZFC has a standard model can be used to prove most theorems (suitably translated into the language of sets) in the mathematics literature
Something else?

REPLY [7 votes]: Only (1).  Unformalized mathematics does not refer to its own consistency or existence of its own models, and its translation into a system like ZFC would not, either.