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 number of "independent universe-sized regions" of space and time is countably infinite

Can you elaborate on that? There are a lot of different kinds of multiverses in physics

As I understand the context of Sean's conversation with Colbert, he was using "multiverse" to refer more or less to "causally disconnected regions of the full spacetime manifold". So he's essentially discussing different regions of space, not any of the more complicated (and likely less countable) versions of what "multiverse" might mean. (I feel like that term gets overused.)

the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers

As long as you believe the holographic principle (and for the most part, I do), there are only a finite number of quantum bits in our Hubble volume anyway.

This is a very fair point. I tried to nod toward related concepts a bit with my comment "assuming space and time are continuous": I think you may get the same reduction of degrees of freedom from uncountable to countable whether you take a holographic approach or just discretize space into Planck-length-sized cells (as I understand it, the big difference is whether the number of qbits scales like area or volume). I'm not even sure that you need all of holography to make the area-law argument: I seem to recall that there's a proof based just on quantum fields in GR that black holes have the highest possible entropy density.

I think I'd need to know more about quantum information theory to be sure whether this really does break my argument, though. (Hmm, the more I think of it, the more I suspect it does.) Perhaps the real lesson from all this is that our (my?) intuitive notion of smooth paths and histories is just wildly inaccurate as a guide to true underlying physics. (Amazingly inaccurate.) I've known that for a while, but I clearly still haven't gotten my brain fully around it.