07 April 2010

An infinite number of infinities


Infinity is an interesting beast. It seems though, that people who have worked with it tend to go mad.

One such mathematician was the German, George Cantor. He defined an infinite number to be one that could be put into a one-to-one correspondence with some part of itself.

One example of this is agreeing that the list of positive integers 1, 2, 3, 4, 5, 6 ... (that dot-dot-dot- means that it go on to infinity) is infinite ...?

(at this point you all nod with me and say "ye-e-e-ess...")

and that the set of square numbers (1, 4, 9, 16, 25, 36 ...) is also infinite ...

(nod, nod, "ye-e-e-ess...")

and you can put them together in one-to-one correspondence ...

1...2...3...4....5....6 ...
.................
1...4...9...16...25...36 ...

and they match up and go on forever ...

(a little less certainly ... "ye-e-e-ess...")

so here we have a set of numbers (positive integers) that can be put into a one-to-one correspondence with a part of itself (the subset that we call "square numbers") and hey presto, because the second set doesn't "run out", we can call them "infinite"!

(everyone go "aaahhh!!" then back away slowly.)

But Cantor, not wanting to stop there, proposed that there isn't just one Infinity. Oh no. Because where would the fun be in that? He proposed the idea that there were an infinite set of infinities. And this is how it came about.

One day in 1874 while on the toilet (I just made that part up), Cantor set out to prove that the number of points on a line were less than the number of points on a plane (or in a space of 3 or more (!) dimensions) but what do you know, he couldn't prove that. Instead he proved the opposite, that the number of points was always the same.

I don't get that. Even he didn't get that, and confessed as much in a letter to German mathematician Richard Dedekind in 1877. But I digress.

All this thinking about those infinite points on a line made him wonder if he could put the infinite series 1, 2, 3 ... into one-to-one correspondence with the infinite points on a line.

Could he do it? Do you think infinity works like that?

Huh?

Duh, no of course not! I am a speech pathologist and even I can see that you can put 1 and 2 on really really really really really reeeally close points on that line but there will ALWAYS be a point in between those where the 2 should have gone!

So he had a think while in the shower one day (I made that bit up too) and decided that the infinite number of points in the line was a greater infinity than the infinite number of positive integers in the series 1, 2, 3 ...

Thus began his cogitations on the possibility of a number of "infinities."

And how many infinities did he propose there were? An infinite number.

Of course.

An infinite number of infinities, called "transfinite" numbers. Just what I need.

And with that, I go off to bed. Good night.

8 comments:

Emily Sue said...

As someone who has a close, loving relationship with words but an uneasy truce with numbers, this is enough to make me cry. I understood all the words but I have no idea what you just said.

On the other hand, if I nodded sagely and said, "Ah yes, that reminds me of Balthazar Bunkum, the 19th century mathematician, who proposed the theory of blahdiblah points in a transfinite circle that rhubarb rhubarb infinity to the power of transfinity (carry the one)..." you'd probably decide we can't be friends after all so maybe it's just as well...

Tracy P. said...

Ahh, the possibilities!

You could totally be a math teacher! Awesome report.

John Ross said...

I was with you right up to transfinities, at which point I metaphorically threw up my hands(or my mind's hands, as it were), and said "to hell with it." Still, a fascinating little post, spicey in a nerdlinger kind of way, bold, yet not too presumptuous, all in all, a fine thing.

Some time try explaining a tesseract to a six year old. And he built a crooked house, was the s.f. short story that came from, can't remember who wrote it though.

veiledturnip said...

Why did I read all of that? You must have a gift - if that were in a book, I would have closed it at the 2nd paragraph!
I see though that your curiosity got the better of you and you chose to read this book first!
'predsk'

~Taz~ said...

hahaha :) too much thinking for me..................... lol :)

Crazy Sister said...

That was fantastic! Laugh-out-loud mathematics.

John Ross said...

Laugh out loud Mathematics indeed. There is a book for tween girls, it's called MATH DOESN'T SUCK.

Lindethiel said...

This appealed to my inner nerd :D