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]

I guess to me, the difference between "the expressible subset of the reals" and "the reals" feels quite artificial, perhaps quite anthropocentric. "Expressibility" feels much more like a philosophical criterion than a mathematical one.

I see what you mean about limits now. I guess my original point was at a lower level than such arguments: I was objecting to the suggestion that in an infinite multiverse there would likely be other near-copies of Steven Colbert in the first place (not to the likelihood of their time slots). As noted, the complexity of the history of the universe that produced our Colbert feels to me like it's likely to be much "bigger" than the number of independent chunks in an infinite universe.

So is the set of all quantum histories a separable space, or is it too big for that? If it's not separable, then there's no point arguing what fraction of the Colberts are on at 11, because the odds of having more than one of him are essentially zero anyway.

[jon-leonard.livejournal.com]

We may need to establish what it takes to be a "Colbert", for that matter. If you divide up a volume of space into roughly atom-size pieces, then specifying what atoms (or gaps) go into each one and how they're bonded together gives you only a finite number of possibilities. So while the question of how many quantum histories there are is interesting, subtracting out stuff like ordinary thermal variation really shrinks the number of possibilities to consider.

So, if you really think there are an infinite number of universe-like regions, and some object of interest is possible, I don't see how to avoid the conclusion that it shows up an infinite number of times. (Infinity times a positive real number is still infinite.)

I'll happily accept that that's not useful, but still...

(Oh, and I do think quantum histories are separable, but maybe I need to learn more quantum to be sure, or even be convinced that it matters.)