# FOM: consis-completeness again

Torkel Franzen torkel at sm.luth.se
Fri Feb 6 05:27:40 EST 1998

```  Mic Detlefsen says:

>Thus, in addition to Steve's question, I am also asking: Are there
>sentences S of PA such that (i) PA does not prove 'S-->Con(T)', (ii) PA |-
>GC--> S, and (iii) if PA |- S, then PA is inconsistent.

But there isn't any counterfactual conditional in this! (iii) is simply
equivalent to "S is unprovable in PA". So S satisfying (i)-(iii) exists
iff GC is not provable in PA, assuming T to refer to some theory for
which Con(T) is not provable in PA. (Take Q to be undecidable in both
PA+not-GC and PA+not-Con(T), and let S be GCvQ.)

>I hope that helps to clarify my question concerning
>consistency-completeness.

No, since the original counterfactual formulation remains to be clarified.

>So, when I assert
>'if G were provable in PA, then PA would be inconsistent', I am not
>asserting the provability of the material version of this conditional in
>either PA or ZF or any other system of mathematical propositions of which I
>am aware.

I have no problem with this, but it is unclear what kind of proof of
the material version will establish the counterfactual statement. Apparently
you do not accept a proof of the form

G is not provable in PA, as we know. Thus, assuming G to be
provable in PA, it follows (by standard propositional logic)
that PA is inconsistent.

>(?*): Ralph, would you still believe in PA's consistency were you to find
>out that G were provable in PA?

>If Ralph were to say 'yes', I would say that he does not have what I would
>count as knowledge of Godel's theorem. He must answer 'no' if I am to give
>him credit for understanding the proof of the theorem. (This is exactly
>what I would say of a student in Ralph's position. Would anyone give a