Ordinals and Godel's Incompleteness theorem

Crispin

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Nov 4, 2017
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I am wondering if there is an official name for the smallest ordibnal — call it O —such that reiteration of the Godel process on some standard axiomatisation for arithmetic O many times results in a set of arithmetical truths that is no longer recursively enumerable (and so no longer 'Godelisable'). Does anyone know? Is this the same as the first non-recursive ordinal?
 
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