[FOM] Godel, Wittgenstein etc.

[Neil Tennant]

On Wed, 7 May 2003 Panu wrote:

> In general, I agree with what Neil says in his posting. 
> Here are just some remarks.
> 
> > If "Cons(F)" is the arithmetized consistency statement, then the
> > turnstile
> > can be that of a possibly very weak subsystem of F---depending, of
> > course,
> > on just how strong F itself is. But the cost of this is a great increase
> > in the length of any proof witnessing the turnstile claim.
> 
> If my memory is not totally failing me, this can be always done in PRA.  
> (or did I misuderstand something?)

I didn't express myself as well as I ought. I should have written
"possibly very much weaker subsystem (than F) of F". That would put the
"depending..." rider in proper context. Apart from PRA, there is the
(weaker) subsystem S_w of Buss. My Mind paper contains the reference.
 
> > So the "real point" that Panu was trying to get at might be put in the
> > following summary way:
> > 
> > ***
> > There is no need for any talk of truth when proving the Godel-sentence
> > G for F. It suffices either 
> > (i) to assume, in meta-F, that F is consistent; or 
> > (ii) to extend F with a reflection principle (but with no new
> > extra-logical vocabulary) so as to be able to prove G.
>  ***
> 
> Right. One might of course add: 
> (iii) to extend F with the arithmetized consitency statement Cons(F)  
> (again, in L(F)). But it is good to see the difference between this and 
> (i). 

Sorry about the omission. Its correction, however, should have been
implicit from what I had said earlier.

> For the present issue, one can restrict the reflection schema to Pi-0-1 
> formulas - and this restriction is equivalent to Cons(F).

Yes, I omitted mention in my email of this restriction of the reflection
schema. But the restriction is explicitly made in my Mind paper. 

Neil Tennant