So assuming that there was a last moment of Socrates's earthly life, and consequently no first moment of his eternal life, we see, further, that, unless the possibility of infinite numbers is granted, it would be quite possible for us logically to doubt the possibility of an eternal life for Socrates on the same grounds as those which led Zeno to assert that motion was impossible and that Achilles could never overtake the Tortoise. If, on the other hand, it be admitted that eternity, at least in the case of Socrates, had a beginning, these same arguments of Zeno would lead any one who denies the possibility of infinite number to conclude that Socrates, like the worm, can never die. Thus is it quite plain that the difficulties about immortality which meet us at the very outset of our inquiry can partly be solved only by the help of the theory of infinite numbers and partly, it would seem, not at all. There is another difficulty about immortality which is quite distinct from this and is analogous to another argument of Zeno. If, indeed, all the instants of time be divided, as before, into the two series of instants at which Socrates was alive and instants at which he was not alive, it follows at once that no instant of time is not accounted for. At none of these instants, however, does Socrates die; obviously he cannot die either when he is alive or when he is dead. Thus it would appear that Socrates never died, and that we ought to re-define the term "mortal" to mean "a human being who is alive at some moments and dead at some." Consequently we must avoid the very tempting conclusion that, because Socrates never died, he was therefore immortal. It is very important carefully to distinguish between the two arguments I have just set forth. The second argument proves quite rigidly that Socrates and, indeed, anybody else, never dies, whether there is or is not a last moment of his life on earth. The first argument proves that, if there is a first moment of Socrates's eternal life, his life on earth never ends. But we have seen that we cannot conclude that this unending life proves that he never is or will be in a state of eternity. FOOTNOTES: [58] By "Dedekind's Axiom," _E. N._, p. 11. [59] _M._, vol. xx., 1910, pp. 134-5. [60] [Here, again, Mr. R*ss*ll's work seems to anticipate some of Mr. Russell's later work, e.g. in _Our Knowledge of the External World as a Field for Scientific Method in Philosophy_, Chicago and London, 1914, pp. 3-4, 55-6, _et passim._--ED.]