Recently I was watching a few very basic lectures on set theory and logic and the following question came up.
Let us have ZFC set theory. The "language" of this set theory is the first order logic. The first order logic is a kind of extension of propositional logic.
When we try to formally define propositional logic, particularly its "syntax", all "valid words" in this "language", we start talking about sets, operations, closed sets, intersections and stuff.
How is it possible to define anything in this circular way at all? To define ZFC we need propositional logic, but to define propositional logic we need ZFC.
Is there any non-circular way of coming from something really basic to set theory formally defining all the entities along the way? Thanks.
