Re: Goedel - interesting problem?

From: Acid Pooh (poohonlsd_at_yahoo.com)
Date: 06/09/04 

Date: 9 Jun 2004 13:43:21 -0700

"|-|erc" <gotcha@beauty.com> wrote in message news:<UEOwc.9665$rz4.7916@news-server.bigpond.net.au>...
> "Daryl McCullough" <daryl@atc-nycorp.com> wrote
> > |-|erc says...
>
> > >You are being selective on what formula you accept.
> > >
> > >G = "this statement has no proof"
> > >~G -> ~"this statement has no proof" -> G has a proof -> G
> > >[contradiction] -> G
> >
> > Your reasoning here is almost correct, except that you
> > left out one assumption. The implication
> >
> > G has a proof -> G
> >
> > is true only if your system is sound (that is, it only proves
> > true sentences). But no system of axioms can *prove* its
> > own soundness. So your proof of G must be done in a different
> > proof system than the one used to formalize "G has a proof".
> >
> > In the case of the Godel statement G for Peano arithmetic,
> > G actually is true, but its truth cannot be proved within
> > Peano arithmetic.
>
>
> you're an exemplary scholar of these notions, but look at the entire load of
> jibberish it forces you to output. what a load of crap, there is nothing
> you are doing here that peano + modus ponens can't do exactly the same.
> in your mass halucinagenic induced state you all summon G as undeniably
> true and concoct local_truthity as its conduit.
>
> I understood godels proof for a decade, there is no use explaining what I now
> understand is a wrong proof to me, if you don't listen to why its wrong and go into
> parrot mode repeatedly I can't help you. G pops up in every such a consistent
> model, I know BIG DEAL. Whats you definition of consistent? It has *a*
> solution either true or false. In those models G will pop up as a valid formula ture,
> and "this statement P is false" will nicely get truncated from the forumula list, added
> to syntactically erronous formula when its syntax is on par with G.
>
> Just think, ALL PROVEN TRUE THINGS HAVE A PROOF.

Duh, this is analytic. If something has been proven to be true, then
it demonstrably has a proof. I hope you don't think that a play on
words is particularly insightful. (Hey, did you hear? All
differentiable things have derivatives!)

At any rate, Godel's incompleteness theorem does not deny this claim.
Very roughly, Godel says that there are sentences in PA that can
neither be proven true nor false in PA. It does NOT say that there is
no proof "absolutely,"--that is, in no extension of PA. If you want a
proof of them (and hence, have a sentence which is proven true, and
hence have a proof), you need to go to a stronger axiomatization.

>
> Why don't you accept that as a true formula? it pops up everywhere too, it craps on G.

G has been proven to be true! It just can't be proved in PA. And,
just for the record, G is not unique: if one is constructing G
number-theoretically, G depends on the enumeration and the
axiomatization one is working with.

'cid 'ooh

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