My Infinity Is Bigger Than Your Infinity

I just read an interesting section about infinity from The Hidden Reality by Brian Green. Scientist who care about this sort of thing are stuck when calculations involve the infinite. Infinite never ends, right? It is the biggest possible–when you tried to one-up your brother and he said, “I can do it a million times better,” and then you said, “Well I can do it infinity-times better,” he was stuck. Even if he tried to say he could do it “double-infinity” times better it didn’t matter because you would just say, there isn’t anything bigger than infinity.Right?

Well try this on: (And I’m paraphrasing this example from the book)

You are offered infinite envelopes, each with an increasing dollar amount–ie. #1 has $1, #2 has $2 and so on for infinity. But then you are given the chance to exchange those envelopes for another set. This time the infinite envelopes have double the dollar amounts, so #1 has $2, #2 has $4, etc. As Green says, at first glance the second set seems better right, but it poses a very weird paradox.

There will be no odd numbers in the second set of envelopes.

So the first set has infinite whole numbers, odd and even. The second set has infinite whole numbers, just even. So, since both are infinite, does that mean that there are actually more even infinite numbers than odd?

If you want to work with the idea of an infinite universe, this makes actual calculations pretty impossible. Distances will be meaningless because you can’t subtract/add/multiply/divide infinity or it wouldn’t be truly infinite.

Someone will just have to come up with a new kind of math, I guess.


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