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Gödel's incompleteness theorems

Gödel proved that, within any axiomatic framework for mathematics there are mathematically true statements that we will never be able to prove are true within that framework."

Source : Marcus du Sautoy, What We Cannot Know: Explorations at the Edge of Knowledge

Gödel developed two theorems dealing with the subject :

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.
The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

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