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.


_________________________________________________________________
The new MSN 8: smart spam protection and 2 months FREE*  
http://join.msn.com/?page=features/junkmail





More information about the cypherpunks-legacy mailing list