Re: Kinds of knowledge was (RE: An elegant distinction (was Re: brz, or Plan B revisited (LONG)))

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.

While it might seem strange to accept whole numbers that can not be
constructed by repeatedly adding one to zero, such moves are not
unheard of. Some models of calculus involve the addition of "numbers"
that are not equal to 0, but are smaller than the difference of any
"ordinary" numbers.

It's also weird to believe that "this statement is false" is true.

Personally, my favorite short form of Goedel's theorem is:

There are unprovable statements in any system powerful enough to prove
anything interesting about strings or numbers.

As with much logic, the definition of interesting is pretty low, so
basically any useful logical system is powerful enough to have such
unprovable statements.

OBconlang, natuaral languages do not operate under the same
constraints and regime as logic. However, the dream of a calculational
system that will decide true or false for any set of statements fed
into it is unattainable. I tend to be skeptical of loglangs as a tool
because people don't seem to use them correctly. I think that this is
fundamental to the way people use logic, but are not logically
consistent themselves.

--
   -- David

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