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]

The distinction between countable and uncountable infinities seems like a distraction: Is there anywhere in physics that you actually need the real numbers? (As opposed to, for example, the expressable subset of the real numbers?)

Oh, constructivism. :P I wouldn't be entirely surprised if physics (or most of it) could get by with just the rationals. For that matter, if at some point space and time turned out to be quantized in a fundamental sense, there might well exist a manifestly countable basis for the space of wavefunctions. But reality certainly feels like R^{3,1} to me... even if that is just years of indoctrination talking. (I like my limits to exist...)

I'm not sure I entirely see the connection between your comment "in terms of limits" and the issues that I'm considering here, but it's possible that we're just paying attention to different aspects of the question.

I'm reluctant to apply much in the way of intuitive reasoning to this sort of problem, though: Human intuition is pretty bad at dealing with rare events (thus the popularity of things like lottery tickets), and this sort of problem really isn't any easier.

I think this one is even less intuitive: at least with lottery tickets we've got some sort of experience dealing with large numbers. Dealing with infinity (and worse, different cardinalities of infinity) is entirely divorced from our intuitive sense. But it really was the nagging suspicion that "the set of all possible histories" might be of much higher cardinality than the integers that got me thinking about this. I'm honestly not sure what that cardinality might be: anything between the reals and goodness knows what. (And as noted, it's even possible that a correct quantum treatment could make the whole thing countable after all.)

As an aside, the set of continuous curves in space is only the same size as the real numbers.

That's a handy fact. Thanks! (I hadn't heard it before, but it feels quite plausible now that I have.)

[marphod.livejournal.com]

There are many places in physics where the rationals are insufficient, and the reals are required. Anything that has a square effect (magnetic field strength, distance covered in an inertial reference frame under acceleration, etc.)

[jon-leonard.livejournal.com]

There are lots of kinds of numbers between the rationals and the reals: Quadratic surds (stuff you get by adding the square root operator), algebraic numbers, etc. Admittedly neither of those are sufficient for physics, since pi keeps showing up, but it's still not clear to me that you need all the reals. Well, all the complex numbers. You can't really do quantum physics with only real numbers.

[steuard.livejournal.com]

Yeah, pi and e are two of the big sticking points that I'd worry about when trying to develop a rationals-only physics. Oh, and by "rationals" I should be clear that I was thinking "complex rationals". (Now I'm starting to worry about whether that's well-defined in the ways I'd need, though.)

Though to be honest, my original thought was actually just that the space-time manifold would be Q^{3,1} rather than R^{3,1}; I hadn't followed that up in my head to conclude that the whole formalism would have to be based only on rationals. I'm still not sure.

[steuard.livejournal.com]

I guess what I'm thinking of here isn't "can you find a closed-form solution to an arbitrary equation of physics with just rationals" but rather "would it be possible to write a fully predictive theory with just the rationals". So I'm picturing less "find the speed of a block sliding down an inclined plane" and more "compute the scattering amplitude for this process in particle physics". And for that, all I really need to have in hand is the Lagrangian of the system (which can usually be written as a polynomial function of position and momentum) and some fancy integration (which I suspect could be sensibly defined over the rationals if one put in some effort).

[jon-leonard.livejournal.com]

Restricting yourself to a countable subset of the reals doesn't mean you have to give up on limits. I'm just making the (pointless?) observation that if you're talking about a specific real number, than your description of it is a finite length string of a finite number of symbols, and there are only countably many such descriptions. Unless you're actually invoking the Axiom of Choice somewhere, the countable/uncountable thing (and the rest of the reals) really shouldn't matter.

However, that's far from the only mathematical tool for dealing with inifinities: Ordinals are often more useful than cardinals, for example. I can make a meaningful claim that a third of the positive integers are divisible by 3, but the cardinal infitities (countable, uncountable: reals, uncountable: real functions, ???) aren't useful in discussing that. It's more a formal claim that in the limit as n goes to infinity, (whole numbers <= n divisible by 3)/(whole numbers <= n) is arbitrarily close to 1/3. I think Sean Carroll's "Maybe more often!" has to be taken in a sense like that.

[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.)

[jon-leonard.livejournal.com]

On the continuous curves thing: A continuous curve can be defined by its values on any dense subset (otherwise it'd be discontinuous!), and a countable product of reals still has the same cardinality as the reals. (And there are countable dense subsets, etc.)

[steuard.livejournal.com]

I like it.