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".That any particular string can be -precisely- defined as truth or false as required by the definition of completeness, is what is not possible.
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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