Lucas can't know when his Gödelization procedure is applicable
For the Lucas argument to work, Lucas must still be able to tell which machines are consistent. But this is mathematically impossible.
G. Lee Bowie (1982).

Note: For further explanation, see Church’s theorem as discussed in Hunter (1973) or Enderton (1972).
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No: computers are limited by Gödel's theorems »No: computers are limited by Gödel's theorems
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The mechanist's dilemma »The mechanist's dilemma
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Lucas can't know when his Gödelization procedure is applicable
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