Empirical evidence for the Axiom of Choice

Empirical evidence for the Axiom of Choice

(Credit to xkcd for the comic, of course.)

Near as I can figure, the go-to framework for mathematics these days is Zermelo-Fraenkel set theory, with or without the Axiom of Choice (denoted as ZF or ZFC, respectively). The Axiom of Choice (AC) was contentious in its day, but the majority of mathematicians now accept it as a valid tool. It is independent of ZF proper, and many useful results require that you either adopt or reject it in your proof, but you're allowed to do either.

The simplified version of AC is that it allows you to take an element from any number of non-empty sets, thereby forming a new set. Where it diverges from ZF is that it allows for rigorous handling of infinite sets, something that is impossible within ZF. Imagine you have an infinite number of buckets, each one containing an infinite number of identical marbles; AC finds it acceptable for you to posit, in the course of a proof, that you somehow select one or more marbles from each of those buckets into a new infinite pile.

Unsurprisingly, this means that AC is equivalent to a number of other famous results, notably the well-ordering principle, which counter-intuitively states that there is a way to totally order any set given the right binary operator, even e.g. %\RR%. Worse yet, it lets you (via the Banach-Tarski paradox) disassemble a tennis ball into five or six pieces and reassemble them into something the size of the sun. In other words, AC is meant to be an abstract mathematical tool rather than a reflection of anything in reality.

Now, I could spend forever and a day discussing this sort of stuff, but let's move along to my thought: what it has to do with the Big Bang.

As you all know, the Big Bang ostensibly kicked off our universe some %13.8% billion years ago. However, some of the details are still kind of hazy, especially the further back you look. Past the chronological blinder of the Planck Epoch—the first %~10^-43% seconds of our universe—there is essentially a giant question mark. All of our physics models and equations break down past that point. Theory says that the big bang, at its moment of inception, was an infinitely dense zero-dimensional point, the mother of all singularities. One moment it's chillin', the next it explodes, and over time becomes our universe.

I don't love this theory, but it's the best fit we've got so far for what we've observed, particularly with red-shifting galaxies and the cosmic microwave background radiation, so we'll run with it for the moment. So where did all of these planets and stars actually come from, originally? There is a long and detailed chronology getting from there to here, but let's concern ourselves with looking at it from an information-theoretic point of view, or rather, 'How did all of this order and structure come out of an infinitesimal point?'

It's unclear, but the leading (though disputed) explanation is inflation, referring to an extremely rapid and sizable burst of expansion in the universe's first moments. There is apparently observational data corroborating this phenomenon, but a lot of the explanation sounds hand-wavy to me, and as though it were made to fit the evidence. The actual large structure of the universe is supposed to have arisen out of scale-invariant quantum fluctuations during this inflation phase, which is a cute notion.

Note, by the way, that entropy was also rapidly increasing during this step. In fact, my gut feeling on the matter is that since entropy is expected to strictly increase until the end of time (maximum entropy), it makes plenty of sense that the Big Bang kernel would have had zero entropy—hell, it's already breaking all the rules. While thermodynamic and information-theoretic entropy are ostensibly two different properties, they're one and the same principle underneath. Unless I'm gravely mistaken, no entropy would translate to complete order, or put another way, absolute uniformity.

If that was indeed the case, its information content may have been nothing more than one infinitely-charged bit (or bits, if you like); and if that were the case, there must have been something between that first nascent moment and the subsequent arrival of complex structure that tore that perfect node of null information asunder. Whether it was indeed quantum fluctuations or some other phenomenon, this is an important point which we will shortly circle back to.

It's far from settled, but a lot of folks in the know believe the universe to be spatially infinite. Our observable universe is currently limited to something just shy of %100% billion light years across; however, necessary but not sufficient for the Big Bang theory is the cosmological principle, which states that we should expect the universe to be overwhelmingly isotropic and homogeneous, particularly on a large scale (%300%+ mil. light years). This is apparently the case, with statistical variance shrinking down to a couple percent or much less, depending on what you're examining.

That last bit is icing on the cake for us. The real victory took place during whatever that moment was where a uniform singularity became perturbed. (Worth noting that if the uniformity thing is true, that mandates that it really was an outside agency that affected it in order to spawn today's superstructure, which of course makes about as little sense as anything else, what with there presumably being no 'outside' from which to act.)

So here's the punchline. If you assume an infinite universe, that means that the energy at one time trapped featureless in that dimensionless point has since been split apart into an infinitude of pieces. "But it's still one universe!" you say. Yes, but I think it's equally valid to recognize that our observable universe is finite, and as such, could be represented by %NN% (if discrete) numbers, or %RR% (if not), or %\aleph_n% if things are crazier than we know. Regardless, it could be described mathematically, as could any other of the infinitely many light cones which probably exist, cozy in their own observable (but creepily similar) universe.

Likewise, we could view each observable universe as a subset of the original Big Bang kernel, since that is from whence everything came. It must be representable as a set of equal or larger cardinality to the power set of all observable universe pockets, and therefore the act of splitting it into these subsets was a physical demonstration of the Axiom of Choice in reality!

I'm not sure what exactly that implies, but I find it awfully spooky. I feel like it either means that some things thought impossible are possible, or that this violation happened when the Big Bang kernel was still in its pre-Planck state; but if that's the case, not only do our physical models break down, our fundamental math would be shown not to apply in that realm either, which means it could have been an anti-logical ineffable maelstrom of quasi-reality which we have no hope of ever reasoning about in a meaningful way.

No life among the stars

On Facebook, in response to this article that's been circulating today for some inscrutable reason, one A. Melaragni said:

Apparently the guy is just talking about our own galaxy; there are BILLIONS AND BILLIONS of galaxies in the universe. Not to mention that even if other life doesn't exist in our galaxy now, that doesn't mean it never did or never will. There is just so much room out there that I think it would be bizarre if life DIDN'T exist elsewhere.

Which got me thinking. In the absence of proof positive one way or another about the existence of life elsewhere or elsewhen, we can still do some perfectly legitimate Bayesian reasoning based on what we've [not] observed, and how the universe seems to work.

While I don't really buy it, I'll concede there are some legitimate reasons to think we might be the only life anywhere. The only other plausible option is that there is an abundance of life: is, was, and will be.

My reasoning is that physics doesn't do one-offs. You don't see a new kind of particle exactly once and never again, you don't see one wildly unique type of stellar body sitting among trillions of others in our galaxy. In general, there are strong mathematical reasons why if anything happens once, it (or something similar enough) will most likely happen twice. And if there's anything less likely than someone happening just once, it's something happening just twice, which would make a mockery of probability.

So, I figure it's probably not just us out here, and if that's the case, it's certainly not going to be just us and one or two other planets of life. Let's consider the two main possibilities.

Let there be life.

In the scenario where intelligent life happens, and where you accept my assertion that there will probably be a lot of it, we can draw some conclusions. Crucially, we are unlikely to be unusual, within the range of all life that eventually takes form. Barring a meddling God, the various salient characteristics of a lifeform—longevity, intelligence, size, high reliance on optical/EM sensory input, overall temperament—are gonna end up distributed as big fat Gaussians, and we're gonna be right near the big fat middle of them for most of these things. Yes, there will be truly alien and bizarre creatures out on the fringes, so that's fun, but it's not us.

This also applies to our timeline of development of technology relative to others. We are one species selected at random from all those who have or will exist. There are some cosmological reasons why we might be one of the earlier ones, but I feel that's heavily outweighed by the implicit probabilistic evidence under discussion; if there are to be a million different life-bearing planets, what are the odds we're the very first to start to get our shit together?

Then of course you run into your Fermi paradox, which on the whole is pretty ominous. Without getting too sidetracked in that, it strongly suggests that either we'll never make it to the stars, or when we do, it will be in a form or mode unrecognizable by present-day us.

To recap:

  • If there's any life besides us, there's probably a shitload of it.
  • If there's a shitload of it, lots of them probably have a huge head start.
  • For whatever reason, they're all gone, unrecognizable, or (at best) undetectable. Without exception. This implies that whatever the attractor at work might ultimately be, it is likely inevitable and undeniable.

Basically, for anyone who dreams of 50s-scifi-style cruising around the galaxy and meeting aliens, give it up. Either there aren't any out there, or there is some overwhelmingly strong reason not to make contact on those sorts of terms, or there'll be some pesky obstacle like inexorable self-annihilation in the way. If we ever do make it to the tipping point where we have the social and engineering infrastructure for interstellar flight, and start doing it up in earnest, I figure that's the nail in the coffin for there being any other life out there. So, the other scenario:

Let there be no life.

In this scenario we are, somehow, a black swan—most likely, there would still be an infinite number of aliens in our infinite universe, but they'd be so negligibly rare as to essentially guarantee none in our light cone, which means they effectively don't exist. There's not a whole lot to say on this other than to point out the silver lining—that the massive, invisible, all-subsuming agent watching us hungrily from behind the curtain of the Fermi paradox would suddenly become a non-issue.

It may be a lonely existence, but this scenario is one in which there's no especially good reason why we can't go exploring all over observable space and spread like cancer. There just won't be all that much more to do out there, at least until we say "okay, fuck it" and seed some new form of life deliberately. I think that, of the two scenarios, this is actually the better deal, as it sidesteps the many and sundry sinister explanations for the current deafening silence.

Note to self: there was something worth exploring there that I skipped by. Given certain kinds of systems governed by a relatively small set of rules but seeded with some level of random initial conditions or ongoing perturbations, is it in fact true that it can be less likely for something to happen twice than to happen once? And if so, how much must a thing have to happen before the probabilities pull even again? I think there could be depth there. Maybe more weight behind %0%, %1%, and %oo% being the only relevant quantities of things. Which is arguably already well established. And %oo% is just a gussied-up %0%.

But I'll go digress.