In 1931, the Austrian logician Kurt Gödel published his incompleteness theorem, a result widely considered one of the greatest intellectual achievements of modern times.
The theorem states that in any reasonable mathematical system there will always be true statements that cannot be proved. The result was a huge shock to the mathematical community, where the prevailing view was an unshakeable optimism about the power and reach of their subject. It had been assumed that maths was “complete”, meaning that all mathematical statements are either provable or refutable. The 25-year-old Gödel demonstrated this was incorrect by constructing a true statement that was not provable. Maths, he announced, has its limits.
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