26/03/2025

In Folha, Viana talks about Russell's Paradox.

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

"From the paradise that Cantor created for us, no one can expel us," said the German mathematician David Hilbert (1862–1943).

Indeed, the set theory of his compatriot Georg Cantor (1845–1918) pointed the way to formulating all of mathematics rigorously from a small number of fundamental ideas, while at the same time shedding new light on the mysterious concept of infinity.

But Cantor's paradise came with its poisoned apples: logical contradictions and a surprising difficulty in understanding even what a set is. And when, in 1901, the Englishman Bertrand Russell (1872–1970) formulated his famous paradox, many people must have felt that the foundations of mathematics were cracking.

Russell then proposed considering the set R to be formed by all sets that are not members of themselves, and only those. For example, the set C of spoons is a member of R, while the set N of "non-spoons" is not. And is R itself a member of itself? The problem is that if R is a member of itself, then, by definition, it is not a member of itself. And if R is not a member of itself, then, again by definition, it is because it is a member of itself. There's no escaping it!

Readers who have followed this column for years will have noticed that this is a sophisticated version of the old paradox of the lying Cretan who claims "I am lying." If he is really lying, then that statement is true, that is, he is telling the truth. And if he is telling the truth, then it is because he is really lying. In either case, we arrive at a contradiction.

But while the Cretan paradox seemed a harmless curiosity, appropriate for impressing friends and acquaintances on the best social occasions, Russell's paradox had important consequences for the logical edifice of mathematics. At the heart of the theory that was supposed to rigorously organize the entire discipline, nestled the serpent of contradiction…

Not everyone was terribly shaken. The Frenchman Henri Poincaré (1854–1912), who had previously made it clear that he considered logic a sterile subject, couldn't resist quipped: "Now it's not (merely) sterile, it generates contradictions." For him, the guiding light of mathematics resides in intuition, not in logic.

In this, he could not be more different from Russell, for whom mathematics consists only of abstract statements of the type "if P is true, then Q is true," in which it does not matter what P and Q mean, nor whether they are true. "Mathematics is that subject in which we never know what we are talking about, nor whether what we are saying is true," he provocatively summarized.

To read the full article, visit the newspaper's website.