Re: Python from Wise Guy's Viewpoint

From: Fergus Henderson (fjh_at_cs.mu.oz.au)
Date: 10/31/03 

Date: Fri, 31 Oct 2003 13:29:18 GMT

Ray Blaak <rAYblaaK@STRIPCAPStelus.net> writes:

>My question is this: why is it the case that "there must be true sentences
>which are not provable"? Given that Tarski himself says "there exists a pair
>of contradictory sentences neither of which is provable", couldn't it be the
>case that the most you can say is "I don't know", i.e., I can't prove things
>either way?

Consider the statements

        P: this sentence is not provable within the formal system.
and
        Q: P is false.

If P was provable within the formal system, then (since we're assuming
that the theory is consistent) it would have to be true, which would
imply that it was NOT provable within the formal system, which would be
a contradiction. Therefore P must not be provable within the formal system.
That of course implies that P is true, and that Q is false.

>In particular, Gödel sentences don't seem to have any truth meaning:

Sure they do. For example, P is true, and Q is false.
We just can't prove that within the formal system. 

-- 
Fergus Henderson <fjh@cs.mu.oz.au>  |  "I have always known that the pursuit
The University of Melbourne         |  of excellence is a lethal habit"
WWW: <http://www.cs.mu.oz.au/~fjh>  |     -- the last words of T. S. Garp.