Questions 3-4 Post Write

3. Sets & WFF
Proposition: The intersection of all good sets, S, of L-strings is exactly equal to WFF.
Proof: We can begin by proving that a WFF, α, is subsets of “good” sets. By definition, atomic formulas like (x = y) and (x Є y) are part of S. The atomic formulas can be built up with finitely many applications to make compound formulas. Based on this idea, suppose ϕ is a WFF and ψ is of the form ¬ϕ. ϕ belongs to all good sets by definition, so ψ does as well. This same argument holds for all the remaining cases (ϕ∧ψ), (ϕ∨ψ), (ϕ→ψ), ∀xϕ and ∃xϕ.
Next, we must also show that any α that belongs to every good set must be a WFF. Every WFF contains the atomic formulas and are closed under the application of the connectives and quantifiers, which is the definition of a good set. Thus, there can be a good set that is made exclusively of a WFF. Hence, the intersection of all good sets is exactly equal to WFF.

4. Set Theory
We are proving the basics of set theory by using set theory. If you use something to prove itself, you have not established that the argument has actual foundations. This could be solved by teaching the origins of set theory. If we knew how the language came to be, we would know how to prove it.