A self-Gödelizing machine can still be out-Gödeled
The Gödelizing operator—to be programmable—must be specified by some finite rule. But in that case, the Gödelizing operator is itself formalisable. The resulting system can be shown to contain a formula that's true but can't be proven in the system.
So the Gödelizing procedure still holds against the self Gödelizing machine.


John Lucas (1961).
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Gödelizing operator can defeat Lucas's argument »Gödelizing operator can defeat Lucas's argument
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A self-Gödelizing machine can still be out-Gödeled
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Informal proof »Informal proof
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John Lucas »John Lucas
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