FOM: Truth of G
Raatikainen Panu A K
Praatikainen at elo.helsinki.fi
Thu Nov 9 09:25:48 EST 2000
On 9 Nov 00, at 9:50, Arnon Avron wrote:
> What the pure proof-theoretical argument demonstrates is not G,
> but Con(T)->G. If one wants to go on and prove G (more accurately,
> that G is *True*, because G cannot be proved in T) then I dont see
> how this can be done without
> a reference to the standard model of T.
RE: Please see my first posting. There I myself emphasized that
the argument begins : ASSUME that T is consistent. One the
derives G. Hence, informally: T is consistent -> G.
(Formalizing this in T, one gets: Cons(T) -> G.)
If one then wants to prove G, it is sufficient to assume ... that T is
consistent, i.e. one cannot derive a contradiction in it. By Modus
Ponens, one gets G. See, no models, standard or not ...
By the way, how on earth it is easier to prove in T, or in any theory,
"G is true" than G ?
One should note that one can Gödelize a highly non-sound theory
(extending Q) which does not even have a standard model - as long
as one assumes it is consistent (in the proof theoretical sense).
My point is that if one seriously assumes the proof-theoretical
consistency of T, one already implicitly assumes (the truth of?) G.
>First, the claim that
> G is true already refers to the standard model (it is not true in all
> models of T).
RE: But on can claim G without any reference to a model. Also, if
one the has just a weak theory of truth (e.g. just the T-scheme),
one can also establish T(G) - again, no standard model. Much
depends on how one understands "Truth". (Note, by the way, that
G is a Pi-0-1 sentence, and that the partial truth definition for Pi-0-1
sentences can be defined in T itself (in fact, by a Pi-0-1 formula).)
The rest of Avron's posting deals with the tricky question: in what
sense does Cons(T) "express" the consistency of T.
I think that dealing adequately this issue would take us far to far ...
I prefer to stick to G and Gödel's first theorem.
I am sorry that I used Cons(T) "uncritically" in my posting. Let us
just use the informal "T is consistent " (does not prove B&-B).
PLEASE NOTE: I did not intend to attack the notion of standard
model of PA - I personally find it well-defined and acceptable.
Further, I think that model-theoretic considerations greatly
illuminate what is really going on in Gödel's theorem.
My point was just to emphasize that Gödel's proof does not as
such assume the notion of the standard model and truth in it -
Gödel took great pains to avoid it, in order to convince even those
who had postivistic, formalistic or finistitic convictions.
Consequently, my aim was to just emphasize that Gödel's proof
should have force even for those who find the whole notion of the
standard model of arithmetic suspectable. There are many who do:
Many claim the notion of actual, completed infinite is nonsensical.
But they presumably accept the proof theoretical notion of
consistency of a theory. And that is all that is needed, the rest can
be carried through in a weak base theory, e.g. in PRA, which is
often considered to commit one only to the notion of potential
infinite. The appeal to the standard model of arithmetic is thus not
necessary.
More information about the FOM
mailing list