[FOM] Godel, Wittgenstein etc.

[praatika@mappi.helsinki.fi]

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?)

> 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). 

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

Best

Panu


Panu Raatikainen

PhD., Docent in Theoretical Philosophy
Fellow, Helsinki Collegium for Advanced Studies
University of Helsinki
 
Address: 
Helsinki Collegium for Advanced Studies
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E-mail: panu.raatikainen at helsinki.fi
 
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