The Mathematician Who Stole Infinity

Added: 2026-05-28

[00:00:00]In 1874, 28-year-old mathematician Gayorg Caner published what seemed like a modest technical paper about sets of numbers, but it was a Trojan horse.

[00:00:11]Hidden inside was an idea that would reshape mathematics. Infinity comes in different sizes.

[00:00:17]>> Caner's ideas were earthshattering.

[00:00:20]People didn't have the expectation that there would be different sizes of infinity. And so, it opened up this whole new realm. We now celebrate Cantor as a genius who just like changed the shape of mathematics.

[00:00:35]>> That is a well a kind of myth.

[00:00:37]>> Newly uncovered letters between Caner and a colleague point to a more complicated reality. An origin story built on a secret and a theft.

[00:00:46]>> I couldn't believe my eyes. Those were the letters that I was looking for.

[00:00:51]>> So, how did a single paper become one of the most important proofs in math? and who was left out to get it published.

[00:01:01]For thousands of years, mathematicians treated infinity like a hall of mirrors.

[00:01:06]Endless but untouchable.

[00:01:08]>> Infinity as a concept was a taboo.

[00:01:11]>> It was a philosophical question.

[00:01:13]>> The only kind of infinity mathematicians would accept had a name, potential infinity. A sequence of numbers could go on forever. But forever was a process, not a destination. The numbers are infinite, yes, but they're only potentially infinite. You can have more and more numbers, as many as you want, but you will never have all of them as a completed totality.

[00:01:38]By 1872, two German mathematicians, Richard Dedic and Gayorg Caner, were beginning to push back against more than 2,000 years of mathematical caution.

[00:01:49]They were independently wrestling with the same idea, constructing the number line. At the time, mathematicians assumed the number line was incomplete because it was full of gaps. Take the stretch between two natural numbers.

[00:02:02]Zoom in and you find infinitely many fractions. But no matter how far you zoom, certain numbers never appear.

[00:02:09]There are real numbers like the square root of two that no fraction is equivalent to. You can get closer and closer without ever precisely landing on it. Dedic wondered, what if you could define a real number by the gap it fills? What if the gap is the number?

[00:02:26]Using a method now called a dedicant cut, he showed how to construct a number line that contains every possible limit.

[00:02:34]Independently, Caner also built a complete and continuous number line.

[00:02:38]>> You have these two guys who are formalizing the the mathematical concept of of the real number line. They were on the same wavelength.

[00:02:45]>> Dedic and Caner's work revealed something unsettling. Infinity wasn't hiding at the far end of the number line. It was there in every interval.

[00:02:56]For the first time, Infinity was a real destination.

[00:02:59]>> So, they exchange papers and then by chance they meet.

[00:03:05]>> In the summer of 1872, Cantor and Dedic met while vacationing in a Swiss village.

[00:03:11]>> Cantor and Dedic, they would make very different impressions if we could see them on the screen.

[00:03:16]>> Canto he was very emotional. He wants to show that he can be a renowned mathematician.

[00:03:23]>> While Dedkin is quite the opposite. He was a quiet man, modest and calm.

[00:03:28]>> Dedic was much older than Cantor. He wasn't a guy who like was crazy to get recognition.

[00:03:35]>> Despite their differences, Cantor and Dedic were united by what they dared to do. Follow the path that led to Infinity.

[00:03:47]In November 1873, Caner wrote to Daki, kicking off a lively mathematical correspondence.

[00:03:54]>> These letters are quite famous. They belong to the most well-known correspondences in mathematics in the 19th century.

[00:04:01]>> This correspondence is key to understanding the origins of infinity.

[00:04:05]Caner's letter survived, but Dedicin's side of the exchange was known only from notes he later wrote. The originals were thought to be lost. We see these questions coming from Cantor and a real exchange of new ideas in mathematics.

[00:04:21]>> Allow me to put a question to you. Caner wrote to dedicant. It has a certain theoretical interest for me, but I cannot answer it myself. Perhaps you can.

[00:04:31]Caner's question was about the collections of mathematical objects called sets. A set can be finite like the numbers 1 through 10 or it can be infinite like the natural numbers which can go on forever.

[00:04:45]To compare the sizes of two sets, mathematicians try to pair them up like ballroom dancers where every element of one set corresponds with a unique element of the other.

[00:04:56]To see this in action, let's compare the even numbers to the natural numbers.

[00:05:01]At first glance, there seem to be half as many evens. But pair them up and every even number finds an exact match with a natural number. Counterintuitive as it seems, the two sets are the same size. Both are what mathematicians call countable.

[00:05:18]>> So counter presents the problem, what happens with the real numbers?

[00:05:22]>> The real numbers include all fractions as well as numbers like the square root of two.

[00:05:28]In his letter to dedic asked a simple but radical question, are the real numbers countable? In other words, can they be placed in onetoone correspondence with the natural numbers?

[00:05:41]According to Dedakin's notes, he wrote back that he didn't know how to prove that, but that he had proved something else. The algebraic numbers, or the numbers that you get as solutions to algebra problems, could be put in one:one correspondence with the natural numbers. In other words, the algebraic numbers are countable.

[00:06:01]>> To find out if the real numbers were countable, Dedic suggested that Caner focus on just a small part of the number line between zero and one.

[00:06:10]>> What Digican told him is that you don't really need to consider all the real number line, only just a little interval. And if that is countable, if you can create a list for that, then it's more than enough. If the real numbers were countable, then you could create a complete list of them and match each one with a natural number with none left over. But Caner showed this was impossible.

[00:06:34]>> Caner proved that for any countable list of real numbers, we can produce a real number that's not on the list.

[00:06:42]>> Caner sent his proof to Dedic who according to his notes improved on it significantly.

[00:06:48]Today, here's how we think about the argument. Assume that we already have a complete list of real numbers between 0 and 1. They can be matched perfectly with the natural numbers. Then look at the first digit of the first real number. The second digit of the second, the third digit of the third, and so on.

[00:07:08]Now let's change each of those digits and combine them to create a new number.

[00:07:13]This new number cannot be anywhere on the list. it differs from the first number in the first decimal place and so on. That means that the real numbers cannot be put in a 1:1 correspondence with the natural numbers. They're uncountable.

[00:07:31]The implication was extraordinary. If the real numbers can't be counted, but the natural numbers can be, then one infinity is larger than the other.

[00:07:41]>> So there's at least these two infinities, and that's quite profound.

[00:07:45]the heavens have opened and we realize that there's this vast new realm to look into and investigate.

[00:07:53]>> If infinity could be measured, then it had to be treated as a legitimate mathematical object. Caner knew that not everyone would be willing to accept that.

[00:08:02]>> He knew it was revolutionary. He knew this can have really deep implications to mathematics.

[00:08:08]>> It was simply not the way ideas were presented in mathematics before.

[00:08:14]There was a big obstacle in the way of publishing these ideas. Leopold Chronicer, a mathematical gatekeeper who was on the editorial board of one of the most prestigious journals in the field.

[00:08:25]>> Chronicer is wholly on the side of the finite. He wants to purify pure mathematics, restricting everything to what you can construct on the basis of the natural numbers. Just imagine you have a very important message you want to share with people and you know you cannot tell anybody because it will just lead to hostility.

[00:08:45]>> Caner would have to find a way through the gates of the establishment. He began to build a mathematical Trojan horse, a paper framed around the countability of the algebraic numbers. First he presented Dedaki's proof nearly word for word. It was the perfect decoy.

[00:09:04]just below it. Caner smuggled in his own argument refined with Dedakin's help, showing that the real numbers cannot be counted.

[00:09:13]Together, the two results pointed to the radical conclusion infinity comes in different sizes.

[00:09:20]>> It is a very short paper. It's barely four pages long.

[00:09:24]>> It's famous because it proves something about the set of real numbers. But the title of the paper doesn't highlight that. It reads on a property of algebraic numbers.

[00:09:35]>> I have no doubt Ker would have preferred to write an article with a title there are different sizes of infinity. Look at this crazy Uh but he couldn't.

[00:09:47]>> Caner's paper was published in Chronicer's journal under Caner's name alone.

[00:09:55]Over the next few decades, the idea from Caner's paper took root in the mathematical establishment, forming the basis of set theory, a framework that would become the unifying language for all of mathematics.

[00:10:07]>> There's this famous quote, "No one shall cast us from the paradox that Caner has created for us."

[00:10:14]>> Today, much of modern mathematics is written in the language of set theory.

[00:10:18]In set theory, you can contemplate essentially any kind of mathematical structure whatsoever. The picture that emerges is this incredibly tall tower of mathematical reality in which more and more sets appear. What set theorists are doing is trying to understand the nature of this hierarchy.

[00:10:40]>> Today, Caner is celebrated as the architect of set theory and as a central figure in the foundations of modern mathematics. The story that we usually tell about set theory is that we owe everything to Caner alone.

[00:10:53]>> It's easier to tell an interesting story with just one single character who is the heroic figure.

[00:11:00]>> Some suspected that Cantor's ideas hadn't emerged in isolation, but the historical record was incomplete.

[00:11:06]>> The letters that Teddican sent to Canto, which would include the proof, um were lost. We thought they were lost forever.

[00:11:16]>> To prove how much Dedakin had shaped Cantor's famous paper, the missing letters would have to be found.

[00:11:24]In 2024, mathematician and journalist Demian Goose began searching for Dakin's letters to Caner.

[00:11:30]>> You find 100 sources and you want to check everything.

[00:11:34]>> The search led him to an archive at the University of Hala in Germany where Caner had held an academic position. So, I went there and they handed over just like a little binder with all the letters. That was just like unbelievable.

[00:11:48]>> There was one dated November 30th, 1873.

[00:11:51]Goose knew immediately what it was.

[00:11:54]>> I was just so excited because I knew this must have the the proof that I was looking for.

[00:11:58]>> It was Dakin's proof that would later appear in Cantor's famous paper without attribution. The set of all algebraic numbers, Dedicin wrote, can be put into a onetoone correspondence with the domain of all natural numbers.

[00:12:13]>> It was just like a shock. With Dedakin's letter in hand, Caner's reply sent just before the paper was published suddenly made more sense. Your remarks, Caner wrote, which I value so highly, were of great assistance to me.

[00:12:28]Canour published a paper, stole his friend's result, and put that that result on the title of his paper. Right.

[00:12:37]Nasty business.

[00:12:38]>> I don't see any way of really letting him go without some kind of accusation of of plagiarism.

[00:12:48]>> The newfound letters offer a glimpse into mathematics as a process shaped by people, their ambitions, their collaborations, and their choices. Math is a collective enterprise. One has to be careful with the ethical sides just like it happens more generally with any scientist.

[00:13:05]>> Goose hopes the letters will reshape how mathematicians tell the story of one of the field's greatest discoveries.

[00:13:12]>> A lot of people are still very emotionally invested in this story about KTO and set theory. there is a reason a valid reason why we acknowledge can counter um but I think it's also more interesting to then tell the the complete story