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Godel's Incompleteness

“It is impossible for a system to prove its consistency.”

Kurt Gödel

 

In addition to being a brilliant mathematician, Kurt Gödel was a great philosopher.

His logic, comparable and compared to that of Aristotle, had a profound impact on the thinking of the twentieth century, a period marked by a deconstruction of classical mathematics, with revisions of the bold and fascinating proposals of Cantor, Dedekind and Frege, courageously taken up by none other than Bertrand Russell, Alfred Whitehead and David Hilbert.

Göedel revolutionized the revolution with his two Incompleteness Theorems.  The first, proposed in 1929, states that for any recursive and consistent axiomatic system, such as one that guarantees the arithmetic of natural numbers, there will always be connected propositions that are impossible to prove or disprove, given the primary condition of origin in the axioms themselves.

The second theorem of 1931 ensures that it is impossible for a system to prove its own consistency.

In terms of the philosophical approach that would allow us to unravel the disconcerting manifestations of quantum mechanics, Niels Bohr's Complementarity Principle, Werner Heisenberg's Uncertainty Principle, and Göedel's Incompleteness Theorems form the holy trinity that, expecting quantum mechanics more optimistically, allow us to create a completely new framework in which determinism, which defines a comfortable yet ideal reality, is replaced by a kind of evolutionary comfort and, consequently, allows us to access a factual reality.

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