Credit: Basic Books / Big Think
Key Takeaways
- Georg Cantor proved that infinity is not merely conceptual but a set of actual, comparable mathematical objects.
- His work showed that some infinities are larger than others, overturning long-held assumptions from Aristotle to Galileo.
- Cantor’s set theory also revealed that dimensionality doesn’t change the size of infinite point sets, fundamentally reshaping our understanding of space and numbers.
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Adapted from The Great Math War: How Three Brilliant Minds Fought for the Foundations of Mathematics by Jason Socrates Bardi. Published by Basic Books. Copyright © 2025. All rights reserved.For thousands of years after Aristotle, the most anyone ever said about infinity was that it was infinite. We could imagine infinity but never actually achieve it. Post-Aristotle, infinity was always idealized, never realized — a philosophical construct at best. As something that could never be reached, infinity could never be treated as a proper mathematical object, most believed. Through the millennia, some of the top philosophical and mathematical minds of their day turned their attention to the concept, pondered it at length, and inevitably gave it up.
[Georg] Cantor will come to understand that better than anyone. […] His work is outstanding, but some of his ideas are way too taboo for many mathematicians. Completing an infinity implies somehow mastering an impossibly long iterative process, like counting every whole number all the way to infinity, including all the really, really large numbers we can’t imagine and don’t even have names for. Count to a googolplex of googols multiplied by many more googols. It can’t possibly be done, many mathematicians think, so best to leave infinity hiding in the bush.
“I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite,” Cantor says.
A drop in the infinite bucket
To understand infinity as Cantor sees it, do this thought experiment: Imagine turning out your pockets and emptying everything in them into a gigantic bucket. While you’re at it, throw your clothes in there as well. And your shoes — everything you’ve got on your person. Now imagine throwing into this bucket every object you touched today, from your coffee mug to your bedside lamp. But don’t stop there. Throw in everything you even looked at today. Add every single thing you thought about, whether real or imagined, from cool dragons to hot doughnuts.
How large would your bucket be? What sort of space would it fill? A stadium? A city? An ocean? The entire planet? What if you filled a bucket like this every single day for your entire life? How large would that collection of 30,000 or so buckets filled every day of a long average life be? What would all those buckets look like together? If you stacked them one on top of the other, would they reach past the Oort cloud? Even if they did, what would your selfish little bucket stack be compared to the entirety of the universe? A drop in the ocean? A heartbeat in a hurricane?
This is where Cantor’s genius shines through. Instead of simply throwing up his hands in despair, he goes about reconciling infinity with his abstract concept of a set. The collection of all those things you touched in a day is but one example of a set — a strictly defined group of sometimes thinly related things.
[While] our minds and our outlooks are grounded in the finite […] set theory allows us to contemplate the scale of things beyond all human scales. And then, using mathematics, to actually touch them.
Cantor began to appreciate this in the 1870s, and after 1883, many others began to appreciate it as well. Cantor’s set theory flips the script in a sense. He shows that infinite sets, including the collection of all whole numbers, are completed infinities. They are actual infinities. You can touch them, in other words, even if they’re not countable in the traditional sense. And you can be touched by them.
Grundlagen Einer Allgemeinen
Because Cantor puts actual infinity within reach, this has strange consequences and starts huge controversies. Why? For one thing, he rigorously proves that there’s more than one type of infinity, some larger than others. And he works out a way to compare them.
The smaller infinite sets, according to his theory, are those equivalent to the collection of all the whole numbers — {1, 2, 3, 4, 5, 6, 7, 8, …}. Cantor proves exactly what Galileo fears: You can have a subset of whole numbers that is equal in its infinite size to its parent set. So the set of all even numbers is exactly the same size as the set of evens and odds combined. What once made Galileo groan now makes Cantor grin.
But this astoundingly nonintuitive result seems unreasonable to many, and for several years leading up to 1883, Cantor’s ideas are routinely criticized. Why?
Think of it from Galileo’s point of view: Imagine taking every possible whole number and placing it in a bucket — a single, completed act. Now imagine doing the same thing with a second bucket. Put just some of the numbers from the first bucket into that second bucket, say all the primes (numbers larger than one divisible only by one and themselves): {2, 3, 5, 7, 11, 13, 17, 19, 23, …}. Or take all odd numbers: {1, 3, 5, 7, 9, 11 …}. Or all evens: {2, 4, 6, 8, 10, 12 …}. Or consider just Fibonacci numbers — the set of numbers invented to describe how rabbits reproduce where each number in the series is the sum of the previous two numbers: {1, 1, 2, 3, 5, 8, 13, 21, …}.
Put all the sets aside in separate buckets and compare them. Which bucket is heaviest? Common sense would say the first. It has all the numbers, so it must be heavier. Galileo desperately wants that to be true.
A picture of Georg Ferdinand Ludwig Philipp Cantor. Cantor developed set theory, today a fundamental branch of mathematics. (Credit: Wikimedia Commons)
This makes sense in an everyday, finite way. If you have a bucket filled with, say, all the hammers in Home Depot, and then you compare that to a subset like all the hammers in Home Depot that are pink (a small subset indeed), common sense would say the whole bucket far outweighs the pink one. And it does. But that’s a mistake based on limited, real-world, finite thinking and is actually incorrect when applied to infinite sets.
Infinite sets are not like finite sets at all. They’re infinite. And in infinity, Achilles catches the turtle. Why? When you compare two completed infinite collections of numbers, one being a subset of the other, they are both equally infinite. In fact, the very definition of a completed, countably infinite or “denumerable” set, as Cantor calls it, is specifically that its members can be matched in correspondence one-to-one with the whole numbers. Cantor not only argues that one infinite subset is exactly equivalent to its infinite parent set — he actually proves it.
This is where things get truly interesting, because Cantor doesn’t stop there. He also shows that not all infinities are equal. There is a larger type of infinity: The set of all decimals or “real numbers.” He says the set of real numbers is uncountable, or “nondenumerable.”
Think of the number line. There is an infinite number of whole numbers on the line, but there is also an infinite number of decimal expansions you can do between any two given numbers: {1, 1.000001, 1.00001, 1.0001, 1.001, 1.01., 1.1 …}. Did I say this is where things get interesting? Really it’s where they get truly bizarre.
Cantor has already challenged what seems like common sense with infinite sets of whole numbers. Next he will violate one of the most basic tenets of geometry held since the dawn of mathematics — and basically challenge reality itself.
“Je le vois, mais je ne le crois pas!”
The naive view of 3D space prior to Cantor assumes that just as there are an infinite number of points on a line, as you expand into more dimensions, the same is also true. A plane is made up of an infinite number of lines, and 3D space is filled with an infinite number of such planes. So which has more points in it: a line, a plane, or 3D space?
The answer seems intuitively obvious. All the infinite points on a single line are nothing compared to the vastness of an entire plane made up of an infinite number of lines. And any single plane falls flat when compared to the infinite stack of planes that would constitute 3D space. It’s almost too stupid to even ask the question Which has more points? The line is greater than the point. The plane is greater than the line. And 3D space is greater than the single, sorry, petty plane. That’s right, isn’t it?
Not with Cantor. And not with set theory.
By the late 1870s, Cantor had reached the strange result that the size of an infinite set of real numbers is independent of dimension. This means that the infinite collection of points in 3D space is exactly equivalent to the infinity of points in a single 2D plane, which is the same again as all the points captured by a single line in that plane.
Again, this makes no sense to the finite mind. How in the world could one line contain as many points as an entire plane? But they do, Cantor says — and he shows it. The line and the plane are the same in that they are equally uncountable. Real number sets are all equally big infinities regardless of whether they’re arrayed in a line, scattered on a plane, or spewed across the vastness of 3D space. And Cantor demonstrates that the set of whole numbers cannot be aligned in one-to-one correspondence with the real numbers. Big infinity is not countable. It’s nondenumerable. And it’s vastly different than denumerable, countable small infinity.
When Cantor first discovers this, he declares, “Je le vois, mais je ne le crois pas!” (I see it, but I don’t believe it).
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