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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 Pete