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Re: Goldbach Restatement

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  • jack@brennen.net
    ... consistent ... undecidable ... its ... a ... found in a ... be a ... false. The nature of GC implies that it may be undecidable, but cannot be *proven* to
    Message 1 of 58 , Oct 30, 2001
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      --- In primenumbers@y..., Jud McCranie <jud.mccranie@m...> wrote:
      > If GC is shown to be undecidable, then the opposite of GC is NOT
      consistent
      > with the axioms - it is contradictory. The fact that it is
      undecidable
      > says that there is no finite proof in number theory for either it or
      its
      > opposite. But if the GC is false, that itself states that there is
      a
      > finite proof that it is false (i.e. a counterexample that can be
      found in a
      > finite number of steps). So if GC is undecidable, a false GC would
      be a
      > contradiction. So if GC is shown to be undecidable, it can't be
      false.

      The nature of GC implies that it may be undecidable, but cannot
      be *proven* to be undecidable.

      Follow this reasoning:

      (1) GC is either decidable or undecidable; also, GC is either
      true or false.

      (2) If GC is false, it is decidable, since a finite counterexample
      exists.

      (3) The converse of (2): If GC is undecidable, it is true.

      (4) Extending (3): If GC can be proven undecidable, it can be
      proven to be true.

      Statement (4) yields a contradiction: if GC is proven to be
      undecidable, it is proven to be true; if it can be proven true,
      it is not undecidable.

      So if GC is truly undecidable, you'll never prove it :-)
    • Jud McCranie
      ... IF (big if ) GC is shown to be undecidable in standard NT then that would mean that it is true, but it would also mean that there is no proof of that it
      Message 58 of 58 , Oct 31, 2001
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        At 02:44 AM 10/31/2001 -0500, Jack Brennen wrote:

        >Then you agree that a proof that GC is undecidable is by its nature
        >also a proof that GC is true. If GC is in fact undecidable, such a
        >proof cannot be allowed to exist - it would decide GC to be true.

        IF (big "if") GC is shown to be undecidable in standard NT then that would
        mean that it is true, but it would also mean that there is no proof of that
        it is true in NT. It would be like Godel's construction of a statement
        that is true, but can't be proven in the system.

        If GC is false then it is decidable (by showing the counterexample in a
        finite number of steps). So if GC is shown to be undecidable in NT then it
        can't be false.

        To look at it a slightly different way, if X is shown to be undecidable in
        a system, that says two things (1) there is no proof of X in the system,
        (2) there is no proof of ~X in the system. So if GC is shown to be
        undecidable, it can't be false since says that there is no counterexample
        whereas a false GC says that there is a counterexample.


        >I don't disagree with that.

        It conflicts with what you said though. To be undecidable, it is only
        necessary that at least one of X or ~X is undecidable. At most one of them
        can be decidable if the problem is to be undecidable (this is called
        partial decidability for the one that is decidable.)




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        | Jud McCranie |
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        | Programming Achieved with Structure, Clarity, And Logic |
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