I listened to one of Nigel Warburton's podcasts on www.philosophybites.com earlier today. These are 15 minute interviews with philosophers on particular topics. The one I selected was with a Professor Adrian Moore from Oxford, on the subject of Infinity.
Having done some maths in my time, I was interested in what Adrian would say on that angle. He mentioned Hilbert's Hotel, where all the rooms are full even though there are an infinite number of them. It's not a problem, though; they can always fit you in by moving whoever is in room 1 into room 2, whoever is in room 2 into room 3, and so on. You can then have room 1.
Unfortunately Adrian didn't have time to mention an even more amazing point : it doesn't matter if you come with an infinite number of friends - they can all have rooms too ! You move whoever is in room 1 into room 2, whoever is in room 2 into room 4, whoever is in room 3 into room 6, and so on, thus vacating all the odd-numbered rooms, and of course there are an infinite number of these.
Another rather interesting fact there was not time for was that there is more than one size of infinity, indeed there is an infinite number, as proved by Cantor back in 1874. The smallest two he identified are known as Aleph-zero and Continuum. As Adrian said, you can take an infinite collection of objects and "count" them by associating each of them with the numbers 1, 2, 3, and so on. What Adrian didn't say was that this applies just to infinite collections of "size" Aleph-zero.
For example, if you take all the rational numbers (numbers which can be expressed as one integer divided by another, such as 2/1, 1/3, 87659234/762508, but not numbers like the square root of 2) and arrange them in a grid where the first row is 1/1, 2/1, 3/1, 4/1 and so on, the second row is 1/2, 2/2, 3/2, 4/2, and so on, with an infinite number of rows and columns. Now you can start at row 1, column 1 and associate that with "1", go one to the right, ie row 1 column 2 and associate that with "2", go diagonally down and to the left, ie row 2 column 1 and associate that with "3" and so on. If you exclude any repeats (eg 1/2 and 2/4), eventually you will "count" any given rational number once, so we know the size is Aleph-zero, because we can do this counting.
However, if we try this same technique to count all the positive real numbers, it does not work. So that means the infinite collection of real numbers is larger than that of the rational numbers. Cantor called this the Continuum and speculated that it might be Aleph-one, the next largest "size".
Incidentally, Adrian apparently wrote an article in Mathematical American, a special online edition of Scientific American, on the subject of the History of Infinity.
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