The Man Who Counted Infinity

“There is a concept which shatters all others and leaves them in disarray. I am not talking about evil which is only limited to ethics, but about infinity.” Even though one might expect these words to have been written down by a mathematician or a scientist, this is not so. Their author is an Argentine writer, Jorge Luis Borges, who succeeded in revealing the very essence of the problem that has intrigued many thinkers before him. Infinity is a concept that has appeared time after time in all kinds of different philosophical, mathematical and physical discussions, but has always been clouded by difficulties and contradictions. People simply can not grasp infinity directly as we are used to do with other concepts, but we have to imagine infinity in an indirect way. Usually, we describe infinity as an endless or limitless sequence, but we quickly come upon problems, as we are entering a dimension where our intuitions can not be completely trusted.

There is no single infinity

Throughout the history, a great number of scholars have thought about the concept of infinity and they have come to many interesting and important conclusions, but the greatest development in the process of understanding this difficult concept had only occurred in the second half of the 19^th century when a mathematician called Georg Cantor (1845-1918) approached the subject of infinity from a completely new angle.

With a couple of simple definitions Cantor succeeded in setting the boundaries of this complicated area of thought, so that he could examine the concept of infinity thoroughly and systematically. To the great amazement of the scientific and philosophical community he soon found out that there is no single infinity, but, in fact, an infinite number of different infinities. He came to this conclusion after careful and accurate consideration and was also able to formally prove all the phases that lead to his findings, so that his ideas have become incorporated into the very core of modern mathematics.

One of the first challenges he had to overcome was defining what infinity actually was. Of course, the simplest way to define infinity was to describe it as having no limits: infinity is what is larger than anything finite. This is were Cantor made an important step forward as he defined something to be infinite, if some of its parts are as big as the whole. This might sound contradictory as we are all intuitively used to finite dimensions whose parts are always smaller than the whole, but as we have already mentioned, when it comes to questions of infinity, intuition does not offer the best answers.

How to compare infinities?

Cantor did not only invent new ways to deal with infinity, he also came up with the well known set theory that children have now been learning in elementary school for decades. When thinking about his new theory, he soon had to face the question of how to compare the size of two sets. He proceeded from the completely intuitive supposition that two sets were of the same size if each element from the first set could be matched-up with exactly one element from the second set. Two groups of children are of the same size if a child from one group can hold hands and pair up with a child from the other group.

Cantor generalized the method of comparison by “holding hands” to infinite sets. Two infinite sets are of equal size if we can match-up each element from one set with an element from the other set. According to this definition, the set of even numbers is just as big as the set of all natural numbers as we can pair up each natural number with an even number simply by multiplying it by two. One holds hands with two, two holds hands with four, three holds hands with six and so on all the way to infinity. As we have matched-up each element from the first set with an element from the second set, both of the sets, in accordance with our definition, are of the same size, even though we could also say that the entire list of natural numbers is twice as large as the list of all even numbers, because natural numbers are made up of both even and odd numbers.

It was with the help of this strange notion that two sets were of the same size even though one is twice as large as the other that Cantor defined the infinite set. An infinite set contains a subset of the same size, when size is defined according to the principle of holding hands. Natural, even and odd numbers thus represent infinite sets.

How to count points on a line?

The question that immediately arises is whether all infinite sets are of the same size. For example, is a set of all points on a line as big as the set of natural numbers? In other words: is it possible to count all the dots on a line? One of Cantor’s important achievements is the proof that it is impossible to count all the dots on a line. Let us have a look at his proof, also called the diagonal argument.

Suppose we are counting points on a line which is one unit long. Each point is matched-up with a number between zero and one. As points fill out all the line because there can be no space between them, otherwise we would not have a line, but a sequence of dots, all of the points on a line can only be matched up with numbers if we use numbers with an infinite decimal representation, called real numbers. Every point on a line which is one unite long can be matched up with a real number between zero and one.

Now imagine we wrote down all of the real numbers one under the other in an endlessly long list on which every line contains one real number and the lines are marked with natural numbers. Cantor proved that we can always find at least one real number missing from every such list of numbers. How come? We simply take away the first decimal from the first real number on the list and change it, then take the second decimal from the second number and change it … If we continued to do so until we came to the end of the list, we would create a new real number different from any number already written on the list.

Coming back to our analogy with the two groups of children that we have compared in size by matching the children up in pairs, we have now proven that pairing up natural numbers with points on a line or real numbers never works. We can always find one real number or point on the line left that has no pair.

Infinities come in infinite numbers

The diagonal argument can also be used to prove that the set of all subsets of a given set is always larger than the set itself. It is the easiest to do so with natural numbers. We write down the subsets of natural numbers as lists of on and off numbers in the set of natural numbers. Odd numbers, for example, are written as {1,0,1,0,1,0 …}, and prime numbers as {1,1,1,0,1,0,1 …}. Now, we arrange these sets into a long list and apply the same argument as before to show that we can always construct another subset of natural numbers that has not been on the list before, simply by taking one element from each subset and changing its value. The set of all subsets of an infinite set is larger than the set itself. Infinite sets come in infinite numbers.

However, a problem already occurs with the smallest infinite sets. We know that natural numbers are the smallest infinite set. It still remains unclear though which is the next bigger infinite set. Is it the set of real numbers or points on a line? Or is there another infinity in between, bigger than natural numbers and smaller than real numbers? Cantor presumed that such a set does not exist, but he was unable to prove it. Many years later, mathematicians solved this question not by finding the answer, but by showing that the question had no answer at all.

Many books about Cantor’s efforts to solve problems concerning infinity also mention his illness which caused him to spend the last years of his life in a mental hospital. Today, he would have been diagnosed with bipolar disorder or manic depression, but at the time most of the patients suffering from this illness were simply labeled insane. Many writers have implied it was actually his work on infinity that drove Cantor over the edge of sanity. This might sound interesting, but his illness and his research are most probably not in a direct causal relationship.