The problem of the absence of contradictions in mathematics arose at the turn of the nineteenth and twentieth centuries, when set theory was discovered.

At the time, there was a fear that a contradiction could be derived from axioms through purely logical conclusions. If this were the case, the entire edifice of mathematics would collapse, since if a contradiction were discovered somewhere, then anything could be proved: any proposition and its opposite.

The unlimited creation of new sets posed problems, and the problem crystallized in the concept of the set that contains as elements all sets that do not contain themselves as an element.

It is known that every set, except the empty set, is composed of elements. These elements can obviously be sets themselves. For example, it is possible to speak of a set whose elements are the sets {0}, {0, 1}, {0, 1, 2} ecc. it is even possible to conceive of a set that contains itself as an element.

Much simpler to conceive are sets that do not contain themselves as an element. All these sets are combined into a single set, and this creates a serious difficulty.

British mathematician and philosopher Bertrand Russell (1872-1970) formulated the dilemma as follows: in a village, there is a barber who shaves only those men who don't shave themselves. The question then becomes: does the barber shave himself or not? If he shaves himself, then he is among those who don't shave themselves, and therefore, according to his rule, he cannot shave. However, if he does not shave himself, then he is among the group he, again according to his rule, must shave. So, he must shave himself too. In both cases, there is a contradiction.

In set theory, the same contradiction appears when one asks whether the set, defined above, of sets that do not contain themselves as an element, contains itself as an element.

The consequence was obvious: the set of all sets that do not contain themselves as an element could not exist. It was therefore necessary to place a limit on the freedom to create sets. Pioneers in this field were Ernst Zermelo and Abraham Fraenkel. The axiomatization of set theory they developed is known as "ZF set theory."

In this way, this particular contradiction was avoided. But does this exclude all possibility of contradiction, or are contradictions still possible? This problem was so important that David Hilbert placed it second among his famous twenty-three problems presented at the International Congress of Mathematicians, held in Paris in 1900.

Subsequently, many mathematicians, led by Hilbert, attempted to prove the absence of contradictions in mathematics or in specific areas of it. However, Kurt Gödel, in his famous essay "On Formally Undecidable Propositions of Principia Mathematica and Related Systems", demonstrated that it is also impossible to prove the absence of contradictions in a mathematical theory within the same theory. At most, it is possible to prove the absence of contradictions in a specific area of mathematics by moving on to a broader area of the discipline. And so on.

Therefore, with the tools of mathematics it is not possible to demonstrate the absence of contradictions in mathematics itself.

References

Albrecht Beutelspacher, Kleines Mathematikum. Die 101 wichtigsten Fragen und Antworten zur Mathematik (Originally published: 15 Nov. 2011)