Tuesday, January 1, 2013

How Many Infinities?

I don't know how old I was when I first encountered "infinity."

How big is it?

Really, really big?

Bigger than the galaxy?


Bigger than the universe?


Bigger than 10 universes?

Yes. Bigger than an infinity of universes?

Whoa! That's big.

Nor do I recall just when someone Cantor's elegant diagonal argument that the set of real numbers is, in some sense, larger than the set of natural numbers. I was probably in college at the time and it was likely either Henry Shapiro or Peter Barnett who took me through the demonstration. So already we're getting sophisticated. We've got sets, we've got natural numbers and real numbers and we've got one-to-one correspondence.

One-to-one correspondence? What's that?

Do you have all your toes?


All your fingers?


Then you should be able to put your toes into one-to-one correspondence with your fingers?

Hmmmm. Let me see. This toe goes with that finger, this next to with that next finger . . . . there, I did it.


But, you know, there's more than one way to skin that cat.

That's fine. What's important is that there's a way to do it.


Now, can you put your, toes, say, into one-to-one correspondence with the hairs on your head?

Of course not.

Well, Cantor proved that you can't put the real numbers into one to one correspondence with the natural numbers.

Hmmm. I think that's going to be tricky.


Well, I've only got just so many toes. Ten fact. And, while I've got many more hairs on my head, it's surely a finite number. But the natural numbers


There's always one more after that, and after that, and so on.


So how can I put the natural numbers into one-to-one correspondence with anything if there's always another one?

And it gets worse with the reals, because there's more of them than of the naturals! That's what Cantor proved.


All of this is just a way of introducing Natalie Anger's New York Times article about infinity. Here's a passage:
“Mathematicians find the concept of infinity so useful, but it can be quite subtle and quite dangerous,” said Ian Stewart, a mathematics researcher at the University of Warwick in England and the author of “Visions of Infinity,” the latest of many books. “If you treat infinity like a normal number, you can come up with all sorts of nonsense, like saying, infinity plus one is equal to infinity, and now we subtract infinity from each side and suddenly naught equals one. You can’t be freewheeling in your use of infinity.” 
Then again, a very different sort of infinity may well be freewheeling you. Based on recent studies of the cosmic microwave afterglow of the Big Bang, with which our known universe began 13.7 billion years ago, many cosmologists now believe that this observable universe is just a tiny, if relentlessly expanding, patch of space-time embedded in a greater universal fabric that is, in a profound sense, infinite. It may be an infinitely large monoverse, or it may be an infinite bubble bath of infinitely budding and inflating multiverses, but infinite it is, and the implications of that infinity are appropriately huge.
If the universe is really infinite, then:
In short, your doppelgängers may be out there and many variants, too, some with much better hair who can play Bach like Glenn Gould. A far less savory thought: There could be a configuration, Dr. Aguirre said, “where the Nazis won the war.”
And so we enter Borges' Library of Babel.

Happy New Year!

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