I am reading a book called ‘A Book of Set Theory’, by Charles C. Pinter. Quite early on he mentions the class axiom, which goes like ‘If S(x) is any statement about an object x, there exists a class which consists of all those elements x which satisfy S(x)’. The author says that this axiom helps us avoid semantic paradoxes. But I don’t get how it does so, because if our S(x) is in fact a semantic statement, then the axiom tells nothing about the existence or non-existence of a class of all elements x satisfying S(x). Such a class may very well exist then.
Just to unpack my silliness clearer, given a statement S(x) in formal language, we can construct a class out of it, but if we have class, as it seems to me, the axiom tells us nothing about whether it has been constructed using a statement in formal language or a semantic one, which thus potentially allows for classes constructed from semantic statements. Where am I wrong?