We don't know that mathematics is consistent
Gödel’s theorem rests on the assumption of a consistent formal theory. But many theories in the history of mathematics have been flawed, and even today the best theories are sometimes called into question.
This indicates that we don't know, but only hope or believe, that our mathematical theories are consistent—and this in turn means that we can't see with certainty the truth of the Gödel sentence.

George Boolos (1990).
 
Note: this argument is directed against Penrose's version of the arguments, rather than Lucas's.
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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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The problem of consistency »The problem of consistency
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We don't know that mathematics is consistent
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The Gödelian insight is all that we need »The Gödelian insight is all that we need
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