09/07/2025

In Folha, Viana explains the 'infinity of infinities'.

First known use of the infinity symbol, by John Wallis – Wikimedia Commons

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

The philosopher Aristotle (384–322 BC) was the first to formulate the distinction between potential infinity and actual infinity. Potential infinity refers to a process that can continue indefinitely, such as the act of counting: 1, 2, 3, … Actual infinity, on the other hand, would be a complete, realized infinity. Aristotle summarily rejected this idea: "Infinity exists in potential, but not in actuality. For it is impossible for infinity to exist in actuality."

In 1656, the Englishman John Wallis published "Arithmetica Infinitorum" (Arithmetic of Infinite Things), a fundamental book in the history of calculus. In it, he used the symbol ∞ to represent infinite quantities. Wallis did not say where he got his inspiration for choosing this symbol, and this became the subject of more or less esoteric (and not very relevant) theories.

It is far more important to understand what infinity is. Sometimes it is treated as a number (as when we write that ∞ + ∞ = ∞), but this is not really correct: ∞ – ∞ makes no sense, for example. And the theory that Georg Cantor (1845–1918) developed two centuries later showed that the situation is much more subtle than previously thought, because there is not a single infinity, there is an infinity of them!

Cantor dared to challenge Aristotle's thinking and treat infinity as an actual object. He started from the age-old idea that two sets have the same cardinality (number of elements) if we can put their elements in one-to-one correspondence. And he discovered that some infinite sets have more elements than others!

Suppose there were as many real numbers as there are natural numbers. Then, the real numbers could all _be listed in sequence: _x₁ , _x₂ , _x₃ , … Consider x = 0 _{, b₁b₂b₃} _… where _b₁ is different from the first decimal digit of _x₁ , _b₂ is different from the first decimal digit of _x₂ , etc. Then x is not in the list, which is absurd because we assumed it contains all the real numbers. This proves that such a list cannot exist.

Cantor called the cardinal number of the set of natural numbers "countable" and the cardinal number of the set of real numbers "continuous." He showed that the countable number is the smallest of all infinite cardinal numbers and, having proven that the continuous number is larger than the countable number, in the way I have described, he went on to exhibit many other cardinal numbers (an infinite hierarchy!) all distinct from and larger than the continuous number.

To complete his theory, he needed to show that the continuum is the second largest of the infinites; in other words, that there is no set with more elements than the natural numbers but fewer than the real numbers. He tried to prove this assertion, which he called the Continuum Hypothesis, for years. Without success.

More than half a century later, Kurt Gödel and Paul Cohen (1934–2007) showed that the Continuum Hypothesis is neither true nor false: it cannot be proven, but it also cannot be excluded ("disproved") from the axioms of mathematics . For this achievement, Cohen won the Fields Medal in 1966.

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