(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.