From: andersw+@pitt.edu (Anders N Weinstein)
Date: 22 Nov 1996 03:48:49 GMT
Organization: University of Pittsburgh

In article <3294EEBB.2CD7@ix.netcom.com>,
Phil Roberts, Jr.  wrote:
>>You are very mistaken.
>>
>>>      Godel's theorem states that in any consistent system
>>>      which is strong enough to produce simple arithmetic
>>>      there are formulae which cannot be proved-in-the-system,
>>>      but which we [standing outside the system] can _see_ to
>>>      be true.
>
>Its important to understand that this is not Lucas's idiosyncratic view of
>Godel.  This _is_ what Godel's theorem amounts to when translated into
>simple English.

I don't see that Godel's theorem says we can _see_ the godel sentence G to
be true. It says that if the system (PM) is consistent then G is 
formally undecidable within the system. That means that the whole infinite
set of PM-theorems contains neither the formula G nor the formula 
consisting of G with a tilde prepended.

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

>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:
>
>	The technical details of Godel's proof itself need not concern us;
>	*no mathematician doubts its soundness* (ibid p. 429).

Again, note Godel's theorem states a *conditional*: 
	
	IF PM-is-consistent THEN  ~PM-theorem('G') & ~PM-theorem(Neg 'G')

One can use the fact that:

	G <-> ~PM-theorem('G')

to conclude

	IF PM-is-consistent THEN G

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).

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).