It is claimed in some logic books that the compactness theorem of first-order logic can be proven using Tychnoff's theorem from topology.
Now, to me this feels very strange because I consider logic as something "lower" in the sense that the theorems of topology follow from the axioms and theorems in logic. How can we avoid circularities then?
Another small question I have is that if we have $A \iff B$ and we prove both directions with contraposition, does there exist a direct proof? You could say we could just follow the steps backwards, but is this always possible?