Ω

Cw: more poetic than usual

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There was a young fellow from Trinity,

Who took the square root of infinity.

But the number of digits,

Gave him the fidgets;

He dropped Math and took up Divinity.

—George Gamow

Lucid lovers me and you
A deal of matchless value
I was always quick to admit defeat
Empty statements of bones and meat
And my heart it shook with fear
I'm a coward behind a shield and spear
Take this sword and throw it far
Let it shine under the morning star

—Kishi Bashi, I am the antichrist to you.

There are about 10⁷⁸ to 10⁸² atoms in the known universe. This number is narrowly beaten out by the 10¹²⁰ possible games of chess. Nearly every chess game, therefore, is one that will never be played, that exists, if anywhere, only in the mind of God or the recesses of some far-away world in the vast multiverse.

Most of these games are bizarre and unintelligible, in no way resembling a way a human would place chess. Many of them involve both sides making totally erratic, crazy moves. The number of chaotic disordered chess moves always outnumbers the number of good ones. It was said that alphazero played chess like an alien, making moves no humans would make or think of. Most chess games would lead us to infer that the player is alien, making bizarre moves no human would make.

Chaos is, in chess, as in life, more plentiful and more abundant than order. Chaos is the default, while order requires structure. In chess, order requires something like us—bipedal apes that evolved on a small rock in an otherwise hostile universe over the course of many billions of years, whose cognitive faculties developed as a cosmic joke, because it happened to be conducive to our reproductive success. It took billions of years of dying and bloodshed and carnage and dinosaurs to get anything like us.

Numbers much bigger than 10¹²⁰ are not easy to find in the universe. But there are much bigger numbers, infinitely many in fact. Contrary to the claims of the ultrafinitists, the numbers just keep going, out past the moon, out past anything you’ll find in the outer recesses of space, beyond anything any creature like us could ever comprehend.

For a while, Skewes’ number was the biggest that proved mathematically relevant, coming at 10^{10^10^34}. It’s since been displaced quite thoroughly by Graham’s number—a number that lies beyond anything that can be reached by exponentiation. As Tim Urban says:

Huge numbers have always both tantalized me and given me nightmares, and until I learned about Graham’s number, I thought the biggest numbers a human could ever conceive of were things like “A googolplex to the googolplexth power,” which would blow my mind when I thought about it. But when I learned about Graham’s number, I realized that not only had I not scratched the surface of a truly huge number, I had been incapable of doing so—I didn’t have the tools. And now that I’ve gained those tools (and you will too today), a googolplex to the googolplexth power sounds like a kid saying “100 plus 100!” when asked to say the biggest number he could think of.

Graham’s number is the upper bound to the solution to some problem. 3↑3=3^3. 3↑↑3=3^3^3. 3↑↑6=3^3^3^3^3^3.

3↑↑↑3=3↑↑3↑↑3.

3↑↑↑↑3=3↑↑↑3↑↑↑3.

Okay, we’ve gotten how the up arrow notation works. If you don’t, just replace the arrows with factorial signs, knowing that the number you’ll get will be effectively zero compared to Graham’s number.

G1=3↑↑↑↑3. That number is already enormous—it’s 3↑↑↑3↑↑↑3 and 3↑↑↑3 was already enormous. 3↑↑↑3↑↑↑3 is too big to contemplate—it’s 3^3^3^3…^3 where the tower of 3s is over 3^3^3^3…^3 , where the tower of 3s is over 7 trillion high. And 3↑↑↑↑3 is a tower of 3s over that high.

But this is just G1. We’re nowhere near Graham’s number. G2=3↑↑↑↑…↑↑↑↑↑3. How many 3s are there? G1! So the arrow notation gives ridiculously huge numbers. Then, to get G2 you make G1 up arrows.

G3=3↑↑↑↑↑↑↑↑↑↑↑↑…↑↑↑↑3, where there are G2 up arrows.

Graham’s number is G64. Googolplex is effectively zero standing next to Graham’s number, as are the other numbers like 10^120. These are child’s play. And yet Graham’s number remains smaller than 100% of numbers. If you were given a number at random and you got Graham’s number you’d be disappointed (that’s true of every number, paradoxically).

The universe has created creatures that can think about things far beyond the universe, that thoroughly outstrip anything the universe will ever contain. We have gazed beyond the universe and witnessed God—witnessed the infinite beyond all that is.

Beyond this, the numbers do not stop. Graham’s number is outstripped by Tree 3—the number of trees one can make from a simple game. You can learn the rules and start drawing trees, as I often did years ago, and still occasionally do when bored—you’ll never draw them all, for they could not fit in the universe. This simple game that anyone can pick up can generate a number of outcomes that far outstrips anything in the universe. And beyond this are other numbers—various Busy Beaver numbers, Loader’s number, Rayo’s number, BIG FOOT, FISH Number 7, LARGE NUMBER GARDEN NUMBER, and various others.

Still, these numbers are 0% of infinity. Even ℵ₀—the smallest infinity—oustrips these numbers by a factor of infinit. Many infinites are bigger; ℵ₁₀, for instance, is bigger than anything we can think of, other than gerrmandered power sets of power sets, or the number of functions of functions of functions of functions. And yet even these infinites are ones we can grasp and move beyond. We can grasp the infinite, the boundless, the endless. We can grasp numbers more than the number of numbers, more than, some think, the number of ways things can be.

God gave man a mind so that man could grasp the mind of God.

Yet we don’t really understand these numbers in any real way. We see them as placeholders, as filling in for a boundless reality. We understand them by their mathematical structure—bigger than some other inconceivable numbers, smaller than others. Yet we understand them much like I understood Green’s theorem when I—at 6 years or so—declared that it’s how you calculate the flux of a vector field. I knew the words to say, for I’d overheard someone saying it, but didn’t know what they meant.

We have a greater understanding of the infinite. But still, we do not understand it in the way that we understand the number five. We cannot picture its physical instantiation in our minds eye, the way we can with five. It’s mathematical properties end up being grotesque, counterintuitive, like some sort of alien God. We discover, bizarrely, that it is as big as part of itself, that the number of numbers between 0 and 1 equals the number of numbers between 0 and Rayo’s number—both ℶ1.

Ω

I am the Alpha and the Omega, the Beginning and the End, the First and the Last.

—Revelation 22:13

You look around and start to cry

Seeing the world’s vast expanse

For ethics under endless skies

Makes change a pathetic, energetic dance

For is not more good better than less?

Is a world pain-filled not superior to one bereft

Of such horrid pain, murder, and theft

But the answer can’t be yes

Infinite tragedies aren’t bad

Unless one does believe

That things are improved by a plan

That just moves people physically

The total amount of goodness doesn’t grow

When people are helped and loved

For the infinite doesn’t know

About health and wealth thereof

In a cosmos just finite

Ethics would be easier

No paradoxes do seem right

The utilitarian’s position is far breezier

I never knew what it was like

To see a conclusion that seems repugnant

Until I heard of endless skies

Where flies die by digits of pi, but what should be our judgment?

They say Bentham was killed

At St. Petersburg or Pasadena

Bentham’s death is just an emblem, still

That rigorous ethics can’t be conceived of

The ultrafinitists think that the infinite doesn’t exist in any real sense. Many other people think that the infinite only exists in the realm of the mathematical, of the theoretical, but not of reality. These people take the infinite to be a grotesque theoretical fantasy, with troubling moral implications, that fortunately we don’t have to think about because it is nowhere in reality.

I remember when I was young, imagining the universe was Graham’s number or Tree 3 units of size. I played imaginary games far later than most children did, up until around 8th grade (though, of course, in later years, in private to avoid ridicule). I imagined the types of aliens that there might be in a universe that big, where every possible physical state is bound to repeat unimaginable numbers of times.

This was something I regarded to be a fantasy. Reality was, I believed, much smaller—containing only around 10^120 particles, perhaps more if there’s a multiverse. Over time, however, I’ve come to believe that reality is infinite in its extent—that the world contains more than Tree 3 or Graham’s number or Loader’s number people—that it contains infinite people.

Why do I think this? In part, it seems that this is the way much of the evidence from physics points. More importantly, however, I accept the self-indication assumption of anthropics. This view says that, given that you exist, you should think more people exist, for if more people exist, it’s likelier any particular person would exist. I’ve argued for this extensively elsewhere.

Most people, after accepting the self-indication assumption, ignore its implications. They treat the fact hat it says you should think the universe is infinite to be an annoying bug to be worked out, rather than something to accept. Yet if we take anthropics seriously—as we should—then we should take the odd implications of SIA seriously.

If you accept the self-indication assumption, then a theory that predicts there are N times as many people as another is N times better, all else equal. More precisely, your existence is evidence that favors a theory N times as well as another theory, if the first theory predicts N times as many people as another. So a theory that predicts there are infinite people will be infinitely better than any finite theory.

But it doesn’t stop there. You shouldn’t just think that there are, say ℵ₁₀ people, for a theory on which there are ℵ₁₁ people is infinitely better, and a theory on which there are ℵ₁₂ people is infinitely better than that. It doens’t stop, it doesn’t culminate in just thinking that the universe is very big—it culminates in thinking that any infinite, of any size, that you can imagine is smaller than the number of real people.

The infinite is no longer just theoretical. It is deeply concrete. There are not just infinite numbers, there are infinite real people that exist. This is what you should accept if you accept the self-indication assumption, a view much more plausible than any of its competitors, one that faces no solid objections yet is supported by ironclad arguments.

How many people should you think there are then? ℵ₁₀ isn’t enough, ℵ₁₀₀ isn’t enough, even ℵ_{Tree 3} isn’t anywhere close to enough. The hierarchy of infinites that mathematicians talk about keeps going forever. An infinite this large takes us beyond the realm of math into the realm of philosophy.

Ω is the symbol for the absolute infinite, the infinite beyond mathematics, beyond reason, beyond human grasp. Cantor created it and identified it with God. And yet it might be real. There cannot be a set of all truths, for creating such a set generates paradox. Perhaps there are Ω truths—and one agent for each truth.

Alexander Pruss has also thought about the infinite goodness, nature, and being of God, the infinite that lies beyond mathematics, beyond sets, beyond human understandings, beyond bijections. He writes:

“God’s value is related to other infinities like (except with a reversal of order) zero is related other infinitesimals. Just as zero is infinitely many times smaller than any other infinitesimal (technically, zero is an infinitesimal—an infinitesimal being a quantity x such that |x| < 1/n for every natural number n), and in an important sense is radically different from them, so too the infinity of God’s value is infinitely many times greater than any other infinity, and in an important sense is radically different from them.”

I do not know if I believe in God. Yet the existence of Ω people gives one a reason to think that there is such a being. For if there is a God, the probability of Ω people being created is much higher than the probability if there is no God.

Ω is the goodness of God. For God can make Ω people—one for each truth. And surely God is better than what he can create, surely his unbounded goodness outstrips the goodness that he might bring about.

I believe that there are Ω people. I believe that there’s a good chance that there exists a being of Ω goodness. I believe that there are more people than numbers than functions of functions of functions of functions. For God’s creation reaches far beyond the expanse of mathematics.

Modality also probably is like this. Pruss has a nice proof that there is no set of all possible worlds. There are Ω possible worlds, then, too many to be part of any set. Whether these worlds are possible or actual, there are Ω of them. Lewis thought that every possible world concretely existed—but he thought there were only ℶ2 of them. In reality, there are more, and not all of them concretely exist. Ω exists, but not every possible world exists.

Is this a good thing? If there is a God, then it seems the answer is clearly yes. For if this is true, contrary to the deranged fantasies of certain Christians and Muslims, reality will be, on the whole, infinitely good for everyone. God has made every soul and will give them all the best life imaginable.

If there is no God, but still there are Ω people, is reality good? I’m inclined to think it is neither good nor bad. That infinities can be compared to other infinites, but they are, in some way, beyond good and evil. That an infinite world cannot be said to be good or bad overall. This was not a conclusion I wanted to accept, but one I was forced to by the numerous paradoxes of the infinite.

The infinite is the most paradoxical area of philosophy. Ironclad arguments seem to point both to the conclusion that it cannot exist and that it does. The evidence from physics spits in the face of our philosophical precepts, laughs at our cries. The boundless universe, unless presided over by the alpha and omega, is beyond axiological evaluation.

For much of my life, I regarded these infinites to be firmly in the realm of fiction. Yet truth is far stranger than fiction. The infinites that seem inconceivable, unimaginable, bizarre end up being not just possible but real. There are Ω people, as real as you or I, out there somewhere.

Numbers beyond conception, beyond math, beyond reason lie out there, waiting.