From: CDJ 
Date: Fri, 22 Nov 1996 10:05:58 -0500
Organization: University of Pittsburgh


On 22 Nov 1996, Anders N Weinstein wrote:

> I don't see that Godel's theorem says we can _see_ the godel sentence G to
> be true.

G's theorem doesn't say a damn thing about our seeing abilities.

> In broad terms, one might say this is a purely syntactic (proof-theoretic) 
> result. 

G's proof was certainly proof-theoretic. But there are numerous ways to
get to the incompleteness result, many of which make heavy use of Tarski.
See, among other things, Boolos and Jeffrey. Smullyan's Incompleteness
book is good too.

> >Its also important to understand that Godel's theorem (and the above simple
> >English statement of that theorem) is all but universally regarded as 
> >valid or "true" by mathematicians:

This person keeps wishy-washing between G's theorem, and G's godel
sentence (i.e., the godel sentence G came up with).

> I think it is correct that almost no mathematician questions the
> soundness of Godel's proof (actually, Wittgenstein once expressed some
> doubts). Indeed all three of the above are representable and provable
> in PM itself (which led to Godel's second incompleteness theorem).

Exactly, G's proof is as unreproachable as any.
 
> But I don't see how Godel's argument could be said to have
> established the consequent G itself nor how we could be said to _see_
> its truth. That is a very different matter than the truth of Godel's
> theorem (the conditional).

Essentially, some people like to sneakily assume that arithmetic is in
fact consistent... (can't you just see! a little addition, a little
multiplication, where's the harm in _that_?)

CDJ