Tuesday, February 25, 2014

A Proof of Self-Proof; 2 of 15


          Every proof uses its own forms of notation, which has its own conveniences and hidden assumptions. I set forth this paper’s notation here in order to simplify critique. To quote the famous logician Humpty Dumpty, “when I use a word, it means just what I choose it to mean – neither more nor less.”
          This paper discusses mathematical propositions, called ‘sentences’. They are defined in terms of each other, using logical connectives such as ‘and’, ‘or’ and ‘no’.  Call logically equivalent sentences ‘equal’; denote equality by ‘=’. Let T denote anything obviously true, and F denote anything obviously false. Therefore T = not F; the former can mean ‘x=x’ or ‘1+1=2’ or ‘0 does not equal 1’; and F denotes the denial of those statements.
          Some sentences are provable. The assertion that the sentence ‘S’ is provable is another sentence; call it “prv S”. To assert prv S is to assert that S is necessarily true, therefore true in all mathematical models.
          Some sentences are true on occasion. Such sentences are ‘possible’. The assertion that the sentence ‘S’ is possible is another sentence; call it “poss S”. To assert poss S is to assert that S is true in some mathematical model; and therefore it cannot be proven false. Therefore poss and prv are conjugate to each other:
          poss S          =        not prv not S
          prv S            =        not poss not S

          The possible is what you can’t prove false; and the provable is what you can’t possibly deny. We can form combinations of these:
          Poss T          =        ‘truth is possible’
          Prv F           =        ‘falsehood is provable’
          Poss F          =        ‘falsehood is possible’
          Prv T           =        ‘truth is provable’

          The last two are easy to simplify. T is provable in all mathematical models, in fact it is the standard of proof; therefore prv T = T. Similarly poss F is false in all mathematical models, therefore poss F = F.
          The statements poss T and prv F involve a deep conundrum; namely, is our proof system itself valid? Perhaps our reasoning methods contain fatal flaws. Is our logic in fact consistent? Does it have a mathematical model? If so, then truth is possible; if not, then falsehood is provable.

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