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From: George Greene on 12 Jun 2010 13:08
On Jun 12, 12:49 pm, George Greene <gree...(a)email.unc.edu> wrote:
> On Jun 12, 11:07 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>
> > Sure it does. Recall Charlie's explanation
>
> >   A wff expresses (represents) the set of numbers that when substituted
> >   for its free variables forms a true (provable) sentence.
>
> Which means that provability is representable WHILE UNPROVABILITY IS
> NOT.
>
> I'm sorry, this is just not acceptable.

Well, maybe it is.
It will turn out that provability is "representable" while
unprovability is merely "expressible".
Maybe this is a difference that you want the terminology to highlight.
Maybe that's why Smullyan did it that way.
But this is only going to be relevant in contexts where there really
is a Standard
model, which is itself a can of worms.




From: George Greene on 12 Jun 2010 13:12
On May 31, 6:30 pm, Rupert <rupertmccal...(a)yahoo.com> wrote:
> On Jun 1, 12:25 am, Charlie-Boo <shymath...(a)gmail.com> wrote:
>
>
>
> > On May 31, 4:11 am, Rupert <rupertmccal...(a)yahoo.com> wrote:
>
> > > On May 30, 10:59 pm, Charlie-Boo <shymath...(a)gmail.com> wrote:
>
> > > > The class of sets represented by PA wffs is the r.e. sets.  We can
> > > > represent no more.
>
> > > > If we add the (true unprovable) Godel sentence G (the wff that
> > > > expresses “It is not provable on itself.” applied to itself) to the
> > > > axioms of PA, which sets can we then represent?  Should that class
> > > > change after adding G to the axioms?  It can’t contain a superclass of
> > > > this class.
>
> > > > C-B
>
> > > Would you be able to clarify exactly what you mean by a set being
> > > "representable" in a theory?
>
> >http://groups.google.com/group/sci.logic/msg/e8946bb14f10372f?hl=en
>
> If a wff with one free variable represents the set of numbers which,
> when substituted into it, yield a true sentence,

Well, IT DOESN'T. Did you read the link??
It Represents the set of numbers which, when substituted into it,
yield a PROVABLE sentence!
It EXPRESSES the set of numbers that yield a true sentence.
But I must complain that it is odd to reintroduce a topic from Years
ago
without re-quoting the definitions.
Simply <producing a link later upon request> is the kind of sloppiness
that
would allow even such a distinguished seeker as Rupert TO GET IT
*WRONG*.
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