"true infinite." His notion of transfinite arithmetic became the standard system for working with infinity within set theory. David Hilbert thought that the role of the actual infinite was relegated only to the abstract realm of mathematics. "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remain for the infinite to play is solely that of an idea." Philosopher William Lane Craig argues that if the past were infinitely long, it would entail the existence of actual infinites in reality. Craig and Sinclair also argue that an actual infinite cannot be formed by successive addition. Quite independent of the absurdities arising from an actual infinite number of past events, the formation of an actual infinite has its own problems. For any finite number n, n+1 equals a finite number. An actual infinity has no immediate predecessor. The Tristram Shandy paradox is an attempt to illustrate the absurdity of an infinite past. Imagine Tristram Shandy, an immortal man who writes his biography so slowly that for every day that he lives, it takes him a year to record that day. Suppose that Shandy had always existed. Since there is a one-to-one correspondence between the number of past days and the number of past years on an infinite past, one could reason that Shandy could write his entire autobiography. From another perspective, Shandy would only get farther and farther behind, and given a past eternity, would be infinitely far behind. Craig asks us to suppose that we met a man who claims to have been counting down from infinity and is now just finishing. We could ask why he did not finish counting yesterday or the day before, since eternity would have been over by then. In fact for any day in the past, if the man would have finished his countdown by day n, he would have finished his countdown by n-1. It follows that the man could not have finished his countdown at any point in the finite past, since he would have already been done. Input from physicists In 1984 physicist Paul Davies deduced a finite-time origin of the universe in a quite different way, from physical grounds: "the universe will eventually die, wallowing, as it were, in its own entropy. This is known among physicists as the 'heat death' of the universe...