  The next obvious task in proof theory, after the proof of the  consistency of arithmetic, was to prove the consistency of analysis,  i.e., of the theory of real numbers. Gentzen did some work in this  direction, but was then assigned to military service in the fall of  1939. (He observed and reported the type, number, and direction of  aircraft that flew above the town of Brunswick, until he was hit by a nervous breakdown in early 1942.) From 1943 on he resumed the work on analysis, but the difficulties intrinsic to the topic were great, as  were the practical difficulties of life caused by the war. Analysis  was to be formulated as a system of second-order arithmetic, which  means that quantification is extended over number-theoretic  predicates or, equivalently, over sets of natural numbers.  Second-order number theory is used in Gentzen's last paper, published in 1943, in which it is briefly shown that the principle of  transfinite induction up to ε0 is derivable in  second-order number theory. 