- --- 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 :-) - 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

IF (big "if") GC is shown to be undecidable in standard NT then that would

>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.

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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