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Re: [PrimeNumbers] Re: Goldbach Restatement

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  • 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 1 of 58 , Oct 31, 2001
      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.)




      +---------------------------------------------------------+
      | Jud McCranie |
      | |
      | Programming Achieved with Structure, Clarity, And Logic |
      +---------------------------------------------------------+
    • 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
        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.)




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