# There are Different Kinds of Infinities

From Brilliance to Madness

This is a story about a brilliant man that became so obsessed with a mathematical question that he went insane and eventually died alone in a sanitarium. This article is also meant as an introduction to the concept of mathematical infinity and the transfinite world.

# Introduction

Think of the natural numbers 1, 2, 3, 4, 5, …. This set is denoted ℕ. It is quite easy to see that this set is infinite because suppose there were only finitely many natural numbers i.e. assume ℕ is a finite set, then we can choose the biggest number and simply add 1 to it. That will generate a new natural number that is not a member of ℕ. But since ℕ was assumed to contain all the natural numbers, we reached a contradiction.

So there are infinitely many natural numbers. However, there are also infinitely many whole numbers, for example, so how do we compare the sizes of these infinite sets?

This question was posed by Germain mathematician Georg Cantor at the end of the 19'th century.

Cantor was born in 1845 in Saint Petersburg, Russia and at the age of eleven, he and his family moved to Germany. The young Cantor was a fantastic musician and a great student. Mathematics came particularly easy to him as he got older and he received his doctorate degree in 1867.

Cantor was not afraid to think outside the box and to challenge the time's wisdom. He began thinking about questions that mathematicians at the time considered almost mathematical heresy.

In particular, Cantor thought of a way of comparing the size (called the cardinality) of infinite sets.

Let’s go back to the question of comparing the cardinality of the natural numbers ℕ by that of the whole numbers ℤ. How do you compare the number of elements of these sets?

The problem of course is that you can’t just count the number of elements in each set and then compare the results. This would work for finite sets, but not for infinite sets because you would never finish counting the elements of the first set, say the natural numbers for example.

Imagine that you have two big piles of small stones and you need to figure out which pile has the greatest number of stones in it, but you are not allowed to count the stones in each pile.

How would you solve this problem?

After thinking about this for a while, you will probably come up with something like the following: You simply remove stones in pairs — one from each pile at a time. The pile that runs out of stones first had the least amount of stones in it to start with.

I think this is a beautiful solution. Note that we don’t know anything about the number of stones in each pile, just which one has the most stones. In particular, if both piles run out of stone at the same time, then there had to be an equal number of stones in the piles.

Cantor comes up with an equivalent method for infinite sets. It is simply this: If there is a one-to-one and onto mapping (called a bijection) between the sets, then their cardinality is the same. A bijection is simply a function

*f: *A →*B*

such that for every element *b* in *B, *there is exactly *one *element *a in *A such that *f(a) = b*. Sometimes we say that the sets are in one-to-one correspondence.

Examples of bijective functions f:ℝ → ℝ include

- f(x) = 2x
- f(x) = x³

for (0, ∞) → (-∞, ∞) we have for example the bijection

- ln(x)

The last example shows that the set of positive real numbers is the same size as all the real numbers!

Note that if the sets are finite then it is exactly the same as comparing them pair-wise as we did in our stone pile example.

# Our Familiar Numbers

Now, it actually turns out that there exists a bijection between ℤ and ℕ. Consider the following function that sends 1 → 1, 2 → -1, 3→ 2, 4→-2, and so on. This map is bijective which shows that the cardinalities of these sets are the same. This might be counter-intuitive since obviously ℕ ⊂ ℤ but in fact the two sets have the same size.

Note that this means that the whole numbers can be listed, i.e. we can put a label on each whole number. The above function gives us the list 1, -1, 2, -2, 3, -3, ….

If there exists a bijection between the natural numbers and a set *A, *we say that *A *is countably infinite (sometimes we call this *countable)*. If an infinite set is not countably infinite, then we call it uncountably infinite (or *uncountable*). By this definition, the set of whole numbers is countably infinite.

How about the cardinality of the rational numbers ℚ? Surely there must be more fractions of whole numbers than natural numbers since all whole numbers are rational, but not vice versa… Right?

Well… Actually, the surprising answer is.. no. The set ℚ is also countable! I will let the task of “finding a bijection f: ℕ → ℚ” be an exercise for the reader.

# Harsh Times

In one of his first papers, Cantor proved that these sets have the same size. One needs to keep in mind that this kind of mathematics had never been done before and the mathematical community wasn’t exactly impressed by his work.

In fact, Kronecker saw Cantor as a “corrupter of youth” for teaching his ideas to the younger generation of mathematicians and was not afraid to spread the word in the community. Unfortunately for Cantor, Kronecker wasn’t the only one with this opinion. Cantor aimed at getting a professorship in Berlin but whenever he applied for a post, he was declined, and it usually involved Kronecker.

In May 1884 Cantor suffered a serious depression. Criticism of his work took its toll on him and at this point in time, he was ready to abandon mathematics for good.

However, Cantor recovered soon after and took on yet another infinite set.

# The Continuum

Cantor now considers the set of real numbers which of course contains all the sets above as subsets. More specifically, he addresses the question “is the set of real numbers countable?”.

As Cantor shows in a paper from 1891, it turns out that the real numbers cannot be put in a one-to-one correspondence with the set of natural numbers i.e. the set of real numbers is uncountably infinite! It is a bigger infinity than that of the natural numbers.

His proof of this is a marvel. A true epiphany of brilliance. Let’s sketch the proof here:

The following is a sketch of proof.

Suppose that the set of real numbers was countable. Then in particular we would have a way to list all the real numbers with 0s and 1s as decimals.

Consider such a list of numbers with 0’s and 1's. I claim that from that (possibly infinite) list, we can always generate a number that is not on that list.

Simply construct a new numbersby taking the first decimal ofsto be the complement of the first decimal of the first number in the list i.e. if it is a 0 it will be a 1 and vice versa. Make the second decimal ofsthe complement of the second decimal of the second number in the list and so on…

It is clear thatscannot be on the list, because it can’t be the first number since it would differ in the first decimal, it can’t be the second number because it would differ in the second decimal, and so on.

This shows that you can’t list all the real numbers.

This proof is called “Cantor’s *diagonal argument”. *Note however that Cantor used infinite binary strings and a couple of details are purposely missed out. But this gives you the idea of how the main argument goes.

It shows that there is more than one kind of infinity, but unfortunately for Cantor, his contemporaries did not appreciate this either.

# Infinitely Many Different Infinities

After this discovery, Cantor had a whole world to explore. A world that no one had seen before. He soon found even bigger infinities than that of the real numbers. In fact, he found infinitely many different infinities!

To introduce some notation, the cardinality of a set is denoted *card(A) *or |A|. The powerset of a set *A *is the set of all the subsets of *A *and is denoted

Note that for a finite set *A*, we have

Remember to include the empty set and the whole set when you calculate the powerset.

If a set *A* has lesser cardinality than a set *B*, we write

In this notation, what Cantor established was

But an amazing result that Cantor proved is that for any set *A*,

This immediately proved the existence of infinitely many different infinities.

He also proved that in fact

i.e. the powerset of the natural numbers has the same cardinality as that of the continuum (the real numbers). He further developed what we now call cardinal arithmetic where one can show for example that

As you can see, Cantor immersed himself in a world where there was always an infinity bigger than the one that you just understood.

# The Continuum Hypothesis

Cantor believed in God and he saw it as his duty to help God give this message to the rest of the world. This would have consequences for Cantor's way of thinking about this as his infinities got bigger and more complicated.

The *aleph *notation ℵ is a list of all infinities in increasing order i.e.

Cantor thought about this list and he now asked himself a critical and scene changing question, namely

Is there an infinity greater than that of the natural numbers but smaller than that of the continuum?

The continuum meaning the real numbers ℝ.

That question is known as *the continuum hypothesis. *To be precise, the continuum hypothesis states

That is, there is no cardinality between the naturals and the reals.

# Into Insanity

Cantor became obsessed with the continuum hypothesis. One moment he thought he had proved it to be true only to realize a mistake and all of a sudden he thought he had proved it to be false. This pattern repeats and for every attempt, he sends out letters to mathematicians, letters that he then has to excuse for a little later when he realized a mistake.

In 1899 Cantor ended up in a sanitorium but recovered again. Shortly after this, however, Cantor’s youngest son Rudolph died suddenly on December 16 that year and this tragedy took a great toll on Cantor. He never really recovered from this and in 1903 he was hospitalized again. Cantor suffered from chronic depression for the rest of his life.

It was a bad cycle, just as he got his energy back he returned to the continuum hypothesis and went mad again and again often with the intention of leaving mathematics for good(he actually took long breaks from math, but always returned). He even began to question God, because he thought that he was part of God’s plan, but then why make it so difficult for him?

Cantor retired in 1913 living in poverty but actually, he achieved some recognition for his work in his own lifetime. One of the greatest mathematicians of the 20’th century David Hilbert famously said:

No one shall expel us from the paradise which Cantor has created for us.

The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home.

Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.

I will end this story with a tribute to Science from the great *William Blake*.

To see a World in a Grain of Sand

And a Heaven in a Wild Flower

Hold Infinity in the palm of your hand

And Eternity in an hour

~ William Blake