Musings on the infinite

[steuard]

A few weeks ago, my friend and former colleague Sean Carroll was a guest on the Colbert Report to promote his book about the nature of time. Toward the end of the interview, they discussed the idea of the "multiverse", which Sean uses to refer to the (possibly) infinite number of "universe-sized" regions within the vast web of space and time where we live. The notion is that if we could somehow travel far enough (faster than light) to regions many times more distant than our telescopes can see, we could find countless independent "universes" that can never talk to each other at all. Some of them would be much like our own but others could be very different, maybe even with different laws of physics. Steven Colbert seemed quite interested:

Colbert: Am I in these other universes?

Carroll: There will be people very much like you.

Colbert: In these other universes, is it possible that my show's on at 11 and John Stewart is at 11:30?

Carroll: Maybe more often!

It's a cute exchange, and it's a variant on the old idea that "in an infinitely big universe, everything that could possibly happen must happen somewhere."

Trouble is, I don't know that I buy that argument, for rather subtle reasons. However we define them, the number of "independent universe-sized regions" of space and time is countably infinite: we could in principle come up with some way of labeling each one by an integer. But many sets (like the real numbers) are uncountably infinite: no matter how you try to label each real number by an integer, you'll miss the vast majority of them. The real numbers are just a much bigger infinity than the integers are. Going on, the set of all possible curves in space is a yet larger infinity. (Assuming space and time are continuous! If they turn out to be discrete, then the set of curves has the same infinite size as the real numbers.)

The thing is, the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers: if anything, it's a still larger infinity. So no matter how large our multiverse may be, it's mathematically impossible for every possible history to occur somewhere. Does that mean that our Steven Colbert (on at 11:30) is the only one? Quite possibly so. I'm not convinced that the multiverse idea opens up as many possibilities as people sometimes think.

Comments

[steuard.livejournal.com]

Oh, blast. Maybe there is a way out, at least sort of, for a reason comparable to the status of the rational numbers in this whole story. You can prove that the rational numbers are countably infinite (by assigning labels based on their integer numerators and denominators, which are countable). But the rationals are also "dense" within the real numbers: if you pick any real number and any allowed deviation (no matter how small, as long as it's not zero), I can find a rational number that's "close enough" to your real.

So even though it's impossible for every single possibility to occur somewhere in the multiverse, perhaps there will always be some region that's "close enough" to any given outcome that you might desire. I don't know if there's a proof of that (I suppose it's possible that the sizes of the infinities involved are too different), but it's at least plausible. Rats.

[beth_leonard]

Maybe 11 is "close enough" to 11:30? ;-)

I've always wondered about the infinite monkeys producing the works of Shakesphere on a typewriter. If you could prove that a monkey's fingers were too large to independently push a single button on the typewriter without pushing another one, or a multitude of other sub-premises about how monkeys type... anyway, I just don't think it's possible even with an infinite number of monkeys and infinite time.

--Beth