for all PA knows, PA might be inconsistent, and there might be an even shorter proof that defect is better.
If the agent checked all proofs of length less than X, and none of them prove statement T, there are no proofs of T shorter than X, even if the theory is inconsistent and T is provable (therefore, by longer proofs).
If the agent checked all proofs of length less than X, and none of them prove statement T, there are no proofs of T shorter than X, even if the theory is inconsistent and T is provable (therefore, by longer proofs).