Re: Kinds of knowledge was (RE: An elegant distinction (was Re: brz, or Plan B revisited (LONG)))
|From:||Peter Bleackley <peter.bleackley@...>|
|Date:||Tuesday, September 27, 2005, 8:22|
Staving David G. Durand:
>On 9/26/05, Peter Bleackley <Peter.Bleackley@...> wrote:
> > Gödel's Incompleteness Theorem states that any system of logic capable of
> > describing itself must either be unable to prove at least one true
> > statement, and therefore be incomplete, or be able to prove at least one
> > false statement, and therefore be inconsistent. This is because the
> > self-describing nature of the system enables you to formulate a proposition
> > equivalent to, "This system of logic cannot prove this statement." If the
> > statement is true, then it cannot be proven, and the system must be
> > incomplete. If not, then the system can prove a falsehood, and therefore
> > contains inconsistencies. Obviously, incomplete systems of logic are
> > preferable to inconsistent ones.
>It's actually a little more subtle than that. The Goedel sentence says
>"There does not exist a number under a certain encoding that
>represents the proof that this sentence is false"
>That number cannot be constructed by any of the proof rules of the
>system. However, it is possible to decide that the statement is false
>and that the system is not incomplete -- but it is only possible by
>adding new "magic numbers" (what are called non-standard extensions of
>the model), which cannot be written down, but whose existence does not
>imply any falsehood in the system.
As I understand it, the sentence should be
"There does not exist a number under a certain encoding that represents the
proof that this statement is true."
Were the statement false, such a number would exist, and therefore the
logical system could prove a falsehood, which is undesirable. For a
self-consistent system, therefore, the statement must be true - and thus
unprovable within the formal system