The man who played with the infinity

By Bogdan Mihailescu

"See this hierarchy? God left it there. I just discovered it."— Georg Cantor (adapted from a statement attributed to him about the order of the universe).

01. The birth of a genius

In the heart of Germany, in a quiet little town, a boy with a brilliant mind was born. Georg Cantor, a precocious child, always fascinated by numbers, shapes and the hidden mysteries of mathematics. From an early age, he asked questions that left adults without answers, questions about the infinite, about things that never end.

In a world obsessed with the finite, Georg Cantor dared to look beyond the limits. Georg Ferdinand Ludwig Philipp Cantor was born on March 3, 1845, in St. Petersburg, into a family of musicians and merchants, in a Europe that saw mathematics as an ordered discipline, an edifice of immutable rules. But his mind was different: a realm where numbers never ended, where infinity was not just a vague concept, but a reality that could be explored, compared, classified.

02. Infinity unleashed

At the University of Berlin, under the guidance of Karl Weierstrass, he began to explore the limits of mathematical analysis. But his heart beat for something deeper:set theoryand the nature of infinity.

Thus he began to work on set theory, a new branch of mathematics that would revolutionize the understanding of infinity as well as the whole of mathematics. No one, not even he, suspected that these studies would open a Pandora's box

03. Hottentots and mathematics.

Since childhood, Cantor was fascinated when he learned about some tribes of Hottentots who had in their vocabulary words like one, two, three and more / more. But they had no words for the number four, five, or beyond. As such these people could not count more than three. Having no money, they practiced barter, exchange in kind. If at the fair a Hottentot wanted to exchange a bunch of plums for a bunch of walnuts, the following problem arose. Are there more nuts or plums? But they could not count plums and walnuts to decide which was more. And then they resorted to the following procedure. They were lining up a walnut on the stall and a plum next to it. And a walnut, and a plum, and so on. If the walnuts ran out and there were plums left, it was clear that the plums were more numerous than the walnuts. Or vice versa, if the plums ran out first, it meant there were fewer of them.

This is how the Hottentots could compare two multitudes that they could not actually count and still decide which was the greater. Then Cantor thought that we can't even count the infinite sets, but we could still compare them by this procedure.

Thus Cantor began to compare the set of natural numbers with the set of even numbers, both of which could not be counted because they were infinite. And he did the same as the Hottentots: He put the natural numbers on one side and the even numbers on the other. So one versus two, two versus four, three versus six, four versus eight, five versus ten, etc. Going to infinity it is seen that every natural number corresponds to an even number. Initially we would expect that the numbers appear to be half of the natural numbers but since both sets are infinite half of infinity is still infinite. But are all infinities equal? Cantor went further. He set about comparing the set of natural numbers with the set of real numbers. Both infinite. So he resorted again to the method of the Hottentots, which he called in a more scientific way "the diagonal argument". (As a quick reminder Real numbers are all numbers on the number line, including natural numbers, but also fractions, decimals and irrational numbers such as π or √2).

And this is how Cantor made the comparison between natural and real numbers as between plums and walnuts?

He imagined a list of all real numbers between zero and one. All these numbers are of the form of a zero followed after the decimal point by sometimes an infinite number of decimals.

Then he imagined a new real number by choosing a digit different from the first digit of the first number in the list, a digit different from the second digit of the second number in the list, and so on.

This new number is different from any other number in the list, so this number is not in the list. This means that you cannot create a list that contains all real numbers. Therefore, the real numbers are infinitely larger than the natural numbers. (more nuts than plums).

So, although we have two infinite sets, the infinity of real numbers is greater than the infinity of natural numbers.

04. An unprepared world.

But Georg Cantor's ideas were too radical for his time. Many mathematicians, accustomed to traditional concepts, were not prepared to accept the existence of several types of infinities. They considered Georg to be a madman, a dangerous dreamer who threatens the foundations of mathematics.

His teachers, including the famous Leopold Kronecker, one of Georg's greatest critics, ridiculed his work. "God made the whole numbers. The rest is the work of man," said Kronecker, denying the idea that infinity can be understood. Leopold Kronecker, accused him of madness and blasphemy, saying that his ideas were an insult to God. Going even further than Leopold, some Christian theologians (especially neo-scholastics) viewed Cantor's work as a challenge to the uniqueness of absolute infinity in the nature of God – at one point equating transfinite number theory with pantheism – an idea which Cantor strongly rejected.

Even the great mathematician Henri Poincaré referred to his ideas as "a serious disease" infecting the discipline of mathematics.

05. Shadows of Madness

The constant criticism, academic isolation and his tireless struggle with ideas that were ahead of his time took a toll on Georg, pushing him into a spiral of suffering. He began to suffer from depression and anxiety, and his mental health gradually deteriorated. He was eventually committed to a sanatorium, where he spent much of his life. Even in moments of madness, Georg kept thinking about the infinite. He wrote articles and papers, trying to explain his ideas and convince others of their truth. But his voice was too weak, and the world was not ready to listen to him.

In the end, everyone's life is finite, even the one who dared to play with the infinite. He died on January 6, 1918, in the sanatorium where he spent the last year of his life.

06. An infinite inheritance

It was only after his death that his ideas began to be accepted and appreciated. Today, set theory is one of the most important branches of mathematics, and Georg Cantor is considered a revolutionary genius. Although rejected during his lifetime, Georg Cantor is now recognized as one of the most revolutionary mathematicians. His set theory became the foundation of modern mathematics, and his exploration of infinity bears his stamp. His contribution to mathematics is much wider than I have presented in this paper. Number theory, trigonometric and ordinal series, set theory, one-to-one correspondences, the Continuum Hypothesis, absolute infinity, the well-ordering theorem, and paradoxes, are just a few of his contributions.

07. Instead of epitaph.

Georg Cantor was buried in the Stadtgottesacker Cemetery in Halle, Germany. At its head is a tombstone on which, instead of an epitaph, an enigmatic equation is engraved:

„C = ℵ₁”

This is an allusion to the Continuum Conjecture, his most beloved conjecture: The power of the continuum (the set of real numbers) is ℵ₁, the next cardinal after ℵ₀. This equation, chosen by Georg himself, is a testimony to his passion for infinity, his obsession to understand it and bring it into the world of mathematics.

08. Conclusion

Georg Cantor showed that infinity is not a static notion, but a dynamic one with a rich and endless structure. Today, the theory of transfinite cardinals is central to logic, topology, and model theory.

He was, indeed, the man who toyed with infinity—and finally beat it.