Ingenious machine's no better than a moronic one
A moronic machine can't extract itself from the Gödel predicament, even if it's given an infinite amount of time. Neither can an ingenious machine extract itself, because it only works faster than the moronic machine.
An ingenious machine may display some mathematical insight—ie it may be able to see shortcuts to proofs—but it still can't recognise the truth of its own Gödel sentence.

Argument anticipated by J. J. C. Smart (1961).
CONTEXT(Help)
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Artificial Intelligence »Artificial Intelligence
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Are thinking computers mathematically possible? [7] »Are thinking computers mathematically possible? [7]
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No: computers are limited by Gödel's theorems »No: computers are limited by Gödel's theorems
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Improved machines »Improved machines
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Ingenious machines could evade the Gödel argument »Ingenious machines could evade the Gödel argument
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Ingenious machine's no better than a moronic one
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Self-reflecting ingenious machine can't be out-Gödeled »Self-reflecting ingenious machine can't be out-Gödeled
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