Unfortunately, the above proof sketch does not work. Mathematics becomes inconsistent if a proposition G is allowed such that G<=>¬⊦G. For a simple proof that

Unfortunately, the above proof sketch does not work. Mathematics becomes inconsistent if a proposition G is allowed such that G<=>¬⊦G. For a simple proof that allowing such a G makes mathematics inconsistent, please see the following: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021