Chapter 1: Setting Off: An Introduction
Every mathematician knows that if 2 + 2 = 5 then Bertrand Russell is the pope. Indeed, Russell is credited with having given a proof of that fact in a lecture by arguing as follows: If 2+2 = 5, then, subtracting 3 from each side, 1 = 2. The pope and Russell are two, therefore they are one. Of course, from the point of view of classical logic, no such proof is needed, since a false statement implies every statement. Contrapositively, every statement implies a given true statement. But suppose we were to take seriously the task of proving that, say, the Four Color Theorem implies that there are infinitely many primes. What are the chances that any of us could come up with a proof that “really uses” the Four Color Theorem? The exercise may seem as pointless as it is difficult, but of course mathematicians do set and perform tasks of this kind on a regular basis. “Use the Bolzano-Weierstrass Theorem to show that if f : [0, 1] → ℝ is continuous, then f is uniformly continuous.” is a typical homework problem in analysis, and the question “Can Chaitin's information-theoretic version of Gödel's First Incompleteness Theorem be used to prove Gödel's Second Incompleteness Theorem?” led to a lovely recent paper by Kritchman and Raz [116]. There is also a well-established practice of showing that a given theorem can be proved without using certain methods, for instance in the exercise of proving the irrationality of 
