A couple of book questions...(one of them about Completeness)
Tyler Durden
camera_lumina at hotmail.com
Mon Dec 2 07:12:52 PST 2002
>That any particular string can be -precisely- defined as truth or false
>as required by the definition of completeness, is what is not possible.
Here we come down to what appears to be at the heart of the confusion as far
as I see it. "True", depending on who's saying it (even in a discussion of
Godelian Completeness), may be different. Mathematical types may define
"true" as being "provably true", meaning something like "this statement can
be derived from the other statements in my system by building up from logic
plus the fundamental axioms".
In Godel, in any formal system there are statements that are true but
unprovable in that system. This would seem to render the notion of "true"
above meaningless. But what it means in a "practical" sense is that there
may be truisms (such as, "there exists no solution to the problem of a^n +
b^n = c^n, where a,b,c and n are integers and n>2"), which are true (and
let's face it, this statement is either true or false) but which can not be
proven given the fundamental axioms of the system. Thus, in order to build
more mathematics with this "truth", it must be incoroprated as an axiom.
(Godel also says that after this "incoporation" is done, there will now be
new unprovable statements.)
I originally mentioned Godel in the context of the notion of the dificulty
of factoring large numbers. My point was that its possible that...
1) Factoring is inherently difficult to do, and no mathematical advances
will ever change that.
and
2) We may never be able to PROVE 1 above.
Thus, we may have to forever live with the uncertainty of the difficulty of
factorization.
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