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Good Math by Mark C. Chu-Carroll

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Going Meta: The Machine That Imitates Machines

Turing machines like the ones we’ve seen are nice, but they’re very limited. If all Turing had done was invent a machine like this, it would have been cool but not really remarkable. The real genius of Turing was his realization that this kind of machine was enough to be able to imitate itself. Turing was able to design a Turing machine whose input tape contained a description of another Turing machine—what we would now call a program. This single machine, known today as a universal Turing machine, could simulate any other Turing machine and therefore could be programmed to perform any computation!

That’s the basic idea of what we call a computer, and it tells us what a computer program really ...

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