Is it impossible that there be two recursive sets T and T* of axioms (in the

Is it impossible that there be two recursive sets T and T* of axioms (in the

Is it impossible that there be two recursive sets T and T* of axioms (in the same language) such that their closures under the same recursive set of recursive rules is identical and yet there is no recursive proof of this fact? It seems impossible but a simple proof of this fact would help elucidate matters!

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