Not All Infinities Are Created Equal

math, and nature, are always waiting to blow your mind

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Did you know that some infinities are larger than others? That infinity is not just this one thing, as we tend to treat it, but is actually made up of many infinities with their own uniquely infinite properties? I never thought about it, honestly, because I’m not a theoretical mathematician, but it turns out it’s true!

When I look around nature, I am frequently reminded of infinity. I am reminded of infinity by many things that perceptually approach infinity, but are actually finite. Countable, if you had nearly infinite days in which to count them. Stars in the sky, grains of sand on the shore, drops in the ocean.

But, as math would have it, infinity becomes more fascinating the more you look at it. Because if you sit down to consider one infinite group, and then you realize that there is another, different group that is also infinite, you soon reach the inescapable conclusion that there must actually be infinite infinities, of varying sizes. ♾️♾️♾️

Here’s the theory.¹ First take one set of numbers, all the natural numbers, which are all of the positive integers. 1, 2, 3, etc. That’s an infinite set of numbers, obviously, just rolling along forever. 1, 2, 3, ♾️

But now, take another other set of numbers. This time, real numbers. Real numbers are all of the numbers, both positive and negative, every fraction, every decimal, everything from one infinite end of the number line to the other. That’s also an infinite set of numbers. -♾️, everything in between, ♾️

And it’s quite obviously a LARGER set of infinite numbers, because it includes our entire infinite set of natural numbers - but still has infinitely more numbers in it besides! ♾️🤯

Turns out, you can do a lot of this, especially if you are a theoretical mathematician, demonstrating how one infinite set is necessarily distinct from or contained in another infinite set, until there are infinite infinities, each discrete, ad infinitum.

And you can be reminded a lot of this, especially if you are a nature-noticer who is amazement-prone, as often as there are stars in the sky or sands on the shore or drops in the ocean - and even moreso.

It’s the mathematics of nature, and the nature of mathematics. It’s just one more way that you can look out your window and have your mind blown in a single moment streaming toward infinity, while infinitely more await.

All of today’s images are from a single moment - sitting in the car - watching a winter sun stream through frosty droplets on the windshield. The droplets are technically countable - but they certainly perceptually approach infinity :)

1. https://www.newscientist.com/article/2459158-mathematicians-have-discovered-a-mind-blowing-new-kind-of-infinity/