Infinity

Watching a recent BBC Horizon programme on infinity, put me in mind of when I was a child and I used to think about infinity a lot. I remember believing that when you went to heaven it was forever. So I'd lie in the field and look up at the infinite depths of the blue sky above and think about being in heaven. This was pleasant for a while as I'd imagine myself sitting up there and it was millions of years in the future and I'm still there. Then it would be millions more years and I'd still be there. And then I'd be overcome with a strange feeling in the pit of my stomach, a sort of queasy sensation which wasn't particularly nice at all. I'd stop thinking about it and then start all over again.

So is infinity real or just a figment of our imagination? Like we can imagine things that aren't real, like unicorns.

But infinity must be real, mustn't it? I mean, just start counting: 1, 2, 3,... infinity. The problem is that it doesn't stop at infinity (or the largest number you can think of). You can simply add 1 more.

Some guy on the Horizon programme said that he wasn't comfortable with infinity and postulated that there was a largest number which when you added 1 to it equalled 0. So you were back where you started. Like starting at a point on the earth's surface and walking in a straight line until you circumnavigated the globe and arrived back where you began. That was an interesting concept but I don't believe it.


But the concept of infinity leads to strange results. Take the apparently simple question of which is bigger, the set of natural numbers (1, 2, 3,...) or the set of even numbers (2, 4, 6,...). The obvious answer is the set of natural numbers. But that is incorrect. They are both the same size. How can this be so? Well, if you believe Georg Cantor, a German mathematician (1845 - 1918) who developed a branch of mathematics called set theory, you have to deal with infinity as a one-to-one correspondence. So taking the two sets of numbers you match 1 (in the first set of natural numbers) with 2 (in the second set of even numbers). Then you match 2 with 4, 3 with 6 and so on. Now for every number in the set of natural numbers you can match it with a number in the even set. Just double it. So the two sets are the same as you can always find an even number to correspond with a natural number.

This leads to all sorts of weirdness with infinity. Take Hilbert's hotel (he was another mathematician from the same era). This is a hotel with an infinity of rooms. You arrive at the hotel and ask for a room only to be told that the hotel is full. But hold on, the manager comes to your rescue and moves all the guests. He puts the guest in Room 1 into Room 2, the guest in Room 2 into Room 3 and so on. Now Room 1 is empty and you get your accommodation after all.

Something even stranger: if we live in an infinite universe, then there must be an infinite number of identical copies of you and indeed everybody else. Now that's a hard one to get your head around.

Look at it this way: let's assume you have many billiard balls and each ball can have only one of 2 colours. That means there are only 16 different ways these balls can be grouped together on a billiard table in a square pattern of 4 balls each. So no matter what combination of 4 balls you choose to place next on the table, it has to be a copy of one of the already existing 16 patterns. Similarly in an infinite universe there are only a certain number of atoms and only a finite way these can be put together. Therefore there has to exist copies of you and everybody else somewhere in the infinite universe. Now that's scary.

Of course, the universe may not be infinite. Besides, the concept of infinity may only exist in our minds.

Fergal
www.uglythump.ie