would have to be false as an arithmetical proposition. Our formal system should not be so badly constructed that it actually allows false propositions to be proved! Thus, it must be the case that there is in fact no proof of Pk(k). But this is precisely what Pk(k) is trying to tell us. What Pk(k) asserts must therefore be a true statement, so Pk(k) must be true as an arithmetical proposition. We have found a true proposition which has no proof within the system! What about its negation ~Pk(k) It follows that we had also better not be able to find a proof of this either. We have just
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