We don't need the entire Lucas arithmetic
A mentalist doesn't have to produce all of the Lucas arithmetic. It's sufficient that he produce enough of the Lucas arithmetic to answer the mechanist at a given step of the game.
The success of the Lucas argument must be evaluated in the context of a particular machine being challenged by particular mentalist.

John Lucas (1970).
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Artificial Intelligence »Artificial Intelligence
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No: computers are limited by Gödel's theorems »No: computers are limited by Gödel's theorems
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Argument from Gödel's theorem is dialectical »Argument from Gödel's theorem is dialectical
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Lucas needs to produce entire Lucas arithmetic »Lucas needs to produce entire Lucas arithmetic
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We don't need the entire Lucas arithmetic
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John Lucas »John Lucas
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