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Topic: Essential Consistency
Replies: 21   Last Post: Mar 25, 2012 1:24 PM

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 Frederick Williams Posts: 2,166 Registered: 10/4/10
Re: Essential Consistency
Posted: Mar 22, 2012 10:28 AM

Zuhair wrote:
>
> On Mar 21, 11:00 pm, MoeBlee <modem...@gmail.com> wrote:

> > On Mar 21, 2:12 pm, Zuhair <zaljo...@gmail.com> wrote:
> >

> > > The idea is that proofs of non standard natural number length are not
> > > real proofs, they are actually not proofs at all, they are only seen
> > > by the theory in question to be proofs, but externally they are
> > > definitely not proofs. I agree it is confusing. I myself need to look
> > > more into that.

> >
> > All you need to say is that every proof is a finite sequence.
> >
> > A proof in system S is a finite sequence of formulas such that each
> > entry in the sequence is an axiom of system S or follows, by an
> > inference rule of system S, from previous entries in the sequence.
> >
> > No need to talk about nonstandard natural numbers in this regard.
> >
> > MoeBlee

>
> You are just not getting the big picture. Yes proof are finite
> sequences of statement etc etc..., we all know that. The idea is that
> a system might recognize some sequences of formulas as proofs, yet
> they are in reality not proofs, those are the kind of proofs that are
> non standard natural number long. This is the whole point. Your point
> is trivial, we already passed what you are asserting.

So you are concerned with two (at least) systems: one T with proofs in
the usual sense, and one T' that takes those proofs (and other things)
as formal objects for investigation? And T' might be wrong about T's
proofs and think that a non standard natural number long sequence of T
formulae is a proof when it isn't?

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting