ABSTRACT
Russell's first major work in logic and foundations was undoubtedly his Principles. Meanwhile, Russell was concentrating on finding a solution to the paradoxes. In the more definitive version of Principia, these difficulties are avoided by taking each number as realized on different levels with infinitely many different copies, and by applying the vicious-circle principle to get different orders of the concept of truth. On a more manageable level, Russell outlines a system of symbolic logic and hints at a type theory as a way of resolving the set-theoretic paradoxes. In the Tractatus, Wittgenstein put forward a number of provocative points about the philosophy of logic and mathematics. Ramsey's remarks are of little mathematical significance because he throws no light on how people might prove by some finite means that the axiom of comprehension is consistent or how they might settle the question of independence of the axiom of choice relative to proofs.
