The Terror of the Singer and the Infinite

2009/11/01 Roa Zubia, Guillermo - Elhuyar Zientzia Iturria: Elhuyar aldizkaria

• Matematika

Mathematics has not often provoked fear. He has chased away many people and enjoys fame of being incomprehensible and boring, but hardly scares him. However, it has happened. The prestigious mathematician Georg Cantor had to overcome fear to maintain research and today is part of the history of mathematics. Object of study, infinite.

Mathematical infinity does not have a very terrifying aspect, it is a symbol in the form of eight lying that rarely appears in calculations. But, beyond the symbol, the concept itself is dark and terrible.

In practice, infinity is nothing terrifying when it arises through a cyclical system. An example: A journey on Earth can be infinite because the same planet does not end anywhere; and another: the two ends of a magnetic tape joined together form a tape that does not end and can be used for continuous video recording (eliminating the previous spin on each tour).

But a non-cyclical infinite escapes from intuition. In short, everything has an end in the world of mortal man. And the concept of infinity does not end. How many numbers are there? Clear, infinite. But what does that mean? It is not an intuitive concept. We know that there are infinite numbers, but we cannot imagine the concept in reality. We must go beyond intuition to begin to understand. Georg Cantor embarked on this journey beyond intuition and died in a psychiatric hospital, full of criticism and philosophical doubts of many mathematicians of the time. But he was a great scientist who opened the way of infinity.

Cantor mathematically defined infinity. It was not enough to know that there are infinite numbers, Cantor showed that this is true, that the set of numbers has infinite elements.

To do this, he analyzed the finite groups. It is easily understood with two sets of three elements. A set of numbers 1, 2 and 3, and a set of numbers 4, 5 and 6, for example. Through a simple operation, a number of the first group and a number of the second can be joined. This operation allows to group all elements by pairs (1 and 4, 2 and 5, 3 and 6). There are no missing or missing elements and there are individual links. Cantor says that proves that both groups have the same number of elements. Doing the same with very large groups, you can know if both have the same number of elements or not, without knowing what it is.

For in the definition of infinity the same. He took the set of natural numbers (1, 2, 3...) and found that it could be individually associated with the set of even numbers (2, 4, 6...). No matter the number of elements, you can make a union by couples, without leftover or missing elements. We know that the second group is the subset of the first and, however, is possible. Cantor indicated that this only occurs with the sets of infinite elements, and thus the infinite was defined.

New infinites infinite

The real revolution came because one step went further: it proposed that not all the infinite are equal. There are infinites larger than others. For example, the set of natural numbers has fewer numbers than the real numbers. The natural numbers are 1, 2, 3, etc., while in the set of real numbers are included the numbers with decimal places. Therefore, the infinite number of real numbers is greater than that of natural numbers.

How many natural numbers are there? For Cantor gave a name to that infinite quantity: Aleph-0 It is a number, but coming from an infinite set, it is a transfinite number. Obviously, Aleph-0 is greater than any natural number, but smaller than other transfinite numbers.

For example, the number of real Aleph-1 numbers is higher. And not only is it greater, but it is an infinite of other kind, since the real numbers cannot be counted, they are not numerous (the natural ones yes).

Cantor was scared. He created a complete arithmetic of transfinite numbers; the next logical step was to investigate infinity, but he thought it came too far. It was at the gates of the religious field, or inside, because the concepts of the infinite and of God have much to see. He received many criticisms and was depressed.

The sad life of Cantor brought a treasure to mathematics. Currently, the arithmetic of transfinite numbers is fundamental in the treatment of infinity. That was what Cantor did great: he looked at mathematical fear for the benefit of the less daring.