02/07/2025

In Folha, Viana talks about Cantor's infinite sets.

Reproduction of Marcelo Viana's column in Folha de São Paulo.

Somewhere, in the early hours of civilization, the shepherd leads the flock to pasture. For each sheep that leaves the pen, he places a pebble in a pile. At the end of the day, he removes one pebble from the pile for each animal that returns to the pen. In this way, he can check that no animal has strayed during the day.

This scene repeats itself countless times, until someone begins to realize that there is something in common between five sheep and five stones, which is the abstract concept of "five." It must have been more or less in this way that we discovered numbers. It can't be a coincidence that the word calculus comes from the Latin calculus , which means pebble or stone.

Abstracting from this age-old idea, we say that two sets have the same cardinal number. (number of elements) if it is possible to establish a one-to-one correspondence between the elements of one and the elements of the other, just as a shepherd does between the flock and the pile. In the second half of the 19th century, the German mathematician Georg Cantor (1845 – 1918) proposed applying this idea even when the sets are infinite. The theory he developed from there brought many surprises, almost paradoxical ones.

To begin, an infinite set can have the same cardinality (the same "number of elements") as a subset of it. For example, the set of integers has the same cardinality as the set of natural numbers (positive integers), even though it also contains negative numbers.

On the other hand, Cantor discovered that the set of real numbers has a cardinality that is strictly larger than that of the set of natural numbers. From this discovery, he constructed a dizzying hierarchy of infinite cardinal numbers, some larger than others. Infinity is not unique; there are an infinity of different infinities!

This theory provoked very diverse, yet intense, reactions. Many rejected it forcefully, mainly because they disagreed with the idea that infinite sets exist as complete and finished things (what philosophers call "actual infinity").

Fellow countryman Leopold Kronecker (1823 – 1891), who had been Cantor's teacher and mentor, was devastating: "Dear sir, your proof seems incomprehensible and even impossible to me. You would make me happy if you abandoned this theory of infinity completely. Actual infinity has no place in mathematics . I have never believed in actual infinity and I see no reason to accept it," he wrote in a letter to Cantor.

Henri Poincaré (1854 – 1912), though more restrained, had no doubt in stating that "Actual infinity does not exist: what we call infinity is merely the endless possibility of creating new objects, regardless of how many objects already exist. Future generations will consider this theory a disease from which they have managed to rid themselves."

Read the full article on the Folha de São Paulo website.