Blackwell Reference Online

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Crudely speaking, the finite is the limited; the infinite is the endless. As history shows, however, such crudeness can be unfortunate. Zeno 's puzzle of how anyone could run a mile if having first to traverse a half-mile, then a quarter-mile, etc., is today often dismissed by saying that infinite sequences can have finite limits. Again, an argument in Kant 's First Antinomy ( Critique of Pure Reason , 1781), that infinite past time could not terminate today, forgets that an infinite line could have one end. On the other hand, space would be finite but unending if it were ‘closed’, curving round and joining up with itself. This ruins another argument of the Antinomy, that any world of finite size would have to be surrounded by infinite emptiness. Other difficulties have likewise succumbed to mathematical advances. For instance al-Ghazali's (1058–1111) stumbling-block, that if time had been flowing for ever then Saturn would have orbited exactly as often as Jupiter instead of only half as often, merely illustrates Cantor 's claim that the numbers in the infinite sequence starting ‘2, 4, 6 …’ can be placed in one-to-one correspondence with (and are in that way ‘just as many as’) those in the sequence starting ‘1, 2, 3, 4, 5, 6 …. ’ Mathematicians readily accept this - together with such oddities as that an infinite hotel with all rooms filled can welcome infinitely many further ... log in or subscribe to read full text