Gödel and Turing: we cannot use classical mathematics to build an interpreter that runs classical mathematics Church and Turing: we can use constructive mathematics to run constructive mathematics Minsky and Turing: we can use constructive mathematics to run classical mathematics
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basic train of thoughts: Gödel's incompleteness theorem (and Turing's subsequent adaptation to computational machinery) shows that mathematics itself is incomplete. that was a shock: mathematics is the domain of all formal languages, but mathematics cannot be generated using them
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there is a small branch of mathematics that only accepts statements as true that have actually been constructed. constructive mathematics turns out to be identical to computation. Church and Turing demonstrated that computation contains itself, i.e. we can compute all computers
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