Hilbert's programme

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continuum hypothesis

Kurt Gödel (1906—1978) Austrian-born American mathematician



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In a famous speech in 1900 the mathematician David Hilbert (1862–1943) identified 23 outstanding problems in mathematics. The first was the continuum hypothesis. The second was the problem of the consistency of mathematics. This evolved into a programme of formalizing mathematical reasoning, with the aim of giving metamathematical proofs of its consistency. (Clearly there is no hope of providing a relative consistency proof of classical mathematics, by giving a model in some other domain. Any domain large and complex enough to provide a model would be raising the same doubts.) The programme was effectively ended by Gödel's theorem of 1931, which showed that any system strong enough to provide a consistency proof of arithmetic would need to make logical and mathematical assumptions at least as strong as arithmetic itself, and hence be just as much prey to hidden inconsistency.

Subjects: Philosophy.

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