Italian version

In 1930, the Austrian logician and mathematician Kurt Gödel, one of the most profound minds of the century, formulated his two incompleteness theorems. The first theorem states that a formal system that includes arithmetic cannot be both consistent—that is, free of contradictions—and complete—that is, capable of proving all the truths of the system. If we do not want to sacrifice consistency, we must accept that there are propositions that cannot be proven—the so-called undecidable propositions.
The liar’s paradox offers us an immediate example of an undecidable proposition. The proposition “This sentence is false”

• It cannot be true, because that would lead to a contradiction.
• It cannot be false, because then it would be true, and we would again have a contradiction.

This type of paradox arises when a proposition is self-referential—that is, when it refers to itself. The core of Gödel’s proof is the construction, within a formal and consistent system that includes arithmetic, of a variant of the liar paradox. To simplify, we can think of the proposition as follows: “This proposition is not provable.” If it is provable, we have a contradiction, and that is impossible in a consistent system. Therefore, it is not provable. We therefore have a true yet unprovable proposition.

This theorem had the effect of a bombshell in the mathematical world. From that moment on, anyone could have stumbled upon—without any way of knowing it in advance—the impossible task of trying to prove an undecidable proposition. For example, can Goldbach’s famous conjecture—“Every even number can be expressed as the sum of two prime numbers”—ever be proven? Or is it one of the unprovable truths?

Our world is permeated by technology based on the principles of quantum physics: microelectronics, lasers, magnetic resonance imaging, LEDs, and the photoelectric effect, which underlies solar energy production. The fascinating aspect of quantum physics is that its large-scale application is based on a theory that is very difficult to understand, full of concepts that contradict common sense, such as the well-known Schrödinger’s cat paradox. This difficulty is summed up by the statement that Richard Feynman, one of the most important physicists of the 20th century, made during a lecture in 1964:
“I think I can safely say that nobody understands quantum mechanics.”

Gödel’s theorem, unlike quantum physics, has no immediate practical implications for everyday life. However, just like the paradoxes of quantum physics, it challenges our perception of an absolute distinction between what is true and false, between what is provable and what is not. The crucial point is that Gödel highlighted an intrinsic limit to the human effort to understand reality. Even within a coherent formal system such as arithmetic, there are truths that cannot be proven even though they lie within the system itself.

One of the most debated implications concerns the nature of the mind. According to Lucas (1961) and Penrose (1989), the human mind is capable of recognizing the truth of undecidable propositions, something a formal machine cannot do. If this insight is correct, the mind would not be reducible to an algorithm: it would possess a non-computational dimension, capable of grasping truths that no formal system can encompass. Gödel himself was a staunch Platonist: he believed that mathematical truths exist independently of the human mind and that mathematics describes an objective, not a constructed, reality.
In this sense, science seems to draw closer to spirituality. To those who legitimately believe only in what is scientifically provable, Gödel reminds us that every system of knowledge, no matter how rigorous, inevitably encounters truths that cannot be explained but only recognized and accepted.

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