“… a grave disease that mathematics will recover from”

This is one of the famous words of the mathematician Henri Poincaré on a theory discussing the foundation of Mathematics. And another man, Leopold Kronecker, who is the professor of the creator of that theory, is the most aggressive critic. Kronecker thought his works were just metaphysical speculations, not Mathematics. He blocked his publications, stopped him from getting university positions, and publicly humiliated him. He called him a charlatan, renegade, and corruptor of youth. Even a few theologians attacked him for the theory of infinites, which looked like it challenged the uniqueness of God.

The man is Georg Cantor — one of the men who inspired me to do mathematics. His Set Theory and hierarchy of infinity are the “problematic” theories he created. Why? Let me explain.

When I was studying in school and the first three years of my degree course, I didn’t know much about infinity. In a few discussions, I was like: infinity is infinity, there is nothing new to explore. During my 4th year even semester, I got to know about Hilbert’s Hotel paradox. That is where I first learned about the properties of infinity. And the Veritasium video about it explains the difference between countable and uncountable infinities using an example.

A bus contains an infinite number of people, and they need to be accommodated in Hilbert’s Hotel, which has an infinite number of rooms numbered 1, 2, 3…. You can accommodate a person by shifting everyone from their room to the next. You can even assign the infinite number of people in the bus to rooms by shifting every person in the hotel to 2 times their room number and assigning the odd numbers to the people in the bus.

But consider the people in the bus whose names contain only A and B and are infinitely long. For example, one person’s name is AABBAABABABABABAABABABA…. Using permutations, you can assign infinitely many people to infinitely many rooms. But there must still be one or more people left out. After assigning each person a room, write the room number and the person in that room as a table. Then take the first letter of the first person’s name and flip it; if it’s A, change it to B. Take the second letter of the second person’s name and flip it. Take the nth letter of the nth person’s name, where n is their room number, and flip it. We can find a new person who didn’t get a room. That is uncountable infinity, which is bigger than countable infinity.

Hilbert’s paradox is a direct illustration of Cantor’s hierarchy of infinities. Cantor also proved this using a similar method. This shatters the old idea that all infinities are the same size. This idea leads to forming algebraic numbers as a sequence of numbers. And he found that algebraic numbers are countable. Then he found every interval has infinitely many numbers which are not part of that sequence, which leads to the discovery of transcendental numbers.

In the 1880s Cantor published his Set Theory, and that discussed infinite sets and uncountable sets. At that time the counter-concepts to Cantor’s theory arose. Kronecker said every mathematical concept should arise from a finite number of steps from natural numbers. For Kronecker, the hierarchy of infinities was not admissible. He even said,

“God made the integers; all else is the work of man.”

He blocked Cantor’s work. He believed Cantor’s ideas on infinity opened doors to paradoxes that would challenge the validity of Mathematics. My mind says he underestimated mathematics — how it loves to solve paradoxes, how paradoxes give joy to mathematicians.

Cantor’s Set Theory argues about the cardinality of sets and discusses different infinities and connects sets through one-to-one correspondence. And it proves the cardinality of integers is smaller than the cardinality of real numbers, and shows it is impossible to make a one-to-one correspondence between integers and real numbers.

I think the great achievement of his concept is that he proved there is a one-to-one correspondence between a line segment, a 2-dimensional square, and a 3-dimensional cube — which means the cardinality of them is the same.

And his ideas on uncountable zero measure sets lead to the new definition of dimension, which allows dimension to be non-integer, called Hausdorff dimension. Dimensions are defined based on space-filling. And his ideas lead to the discovery of the Cantor set, which has dimension approximately 0.63093.

Read our article – Evolution of Dimensions.

Cantor also introduced the well-ordering principle, claiming every set can be well-ordered, and called this a “law of thought.” Many people worked to prove the well-ordering theorem. Later, in 1910, Zermelo proved it. What Cantor tried to do was eliminate paradoxes in mathematics by building a pure axiomatic environment. Even now people are trying to build this kind of environment for Mathematics.

Being Cantor is very hard. He was directly attacked by mathematicians for his theories, which changed the perspective of Mathematics. But later Cantor’s theories were well accepted and celebrated.

“No one shall expel us from the paradise that Cantor has created.”

— David Hilbert

Émile Borel called Cantor’s Set Theory

“An indispensable foundation for modern analysis.”

Felix Hausdorff viewed Cantor as

“The builder of the infinite landscape.”

Bertrand Russell said Cantor’s work was

“One of the greatest achievements of human intelligence.”

Ernst Zermelo considered Cantor’s work

“The natural starting point for the foundations of Mathematics.”

Gödel called Cantor’s vision of the infinite

“Magnificent and unparalleled.”

Yeah! He and his works deserve this praise. Even I feel the same for Cantor’s theories.
But being Cantor is very hard. He faced financial hardships. He was alone. Mental struggles. He was mocked and humiliated by his own professor and colleagues. In one letter to Dedekind he wrote:

“I cannot go on. I fear my ideas will die with me.”

He stopped doing mathematics. He spent days in psychiatric hospitals, battled malnutrition. His life was very painful. But he is great, I was impressed by his ideas. He helped Mathematics strengthen its foundations. His theories are not the disease that came to ruin Mathematics; his theories actually helped to strengthen the foundation of Mathematics.