FOM: Reply to Detlefsen on Consistency-Completeness

Shipman, Joe x2845 shipman at
Wed Feb 11 10:20:31 EST 1998

Dear Mic,
I was not claiming that PA was omega-inconsistent!  Rather, I was trying to 
come as close as possible to a *mathematically formalizable* version 
of your query.  You admit there seems to be a difficulty in 
formalizing it without trivializing it in some way, and I agree that this is an 
interesting philosophical issue, but some of us don't see a way to 
make your query less vague without making it formalizable.  Can you 
please clarify whether there are any technical questions you are 
trying to resolve (by providing a formalizable statement of them)?  
If the technical situation is clear to you and your query is purely philosophical
that is OK, Ph.o'M is not out of place on F.o'M, but at least those 
of us less comfortable with unformalizable questions can safely 
ignore it.
Joe Shipman 

> (#)  If S is provable in PA, then PA is inconsistent
> Consider the instance of (#) obtained by substituting neg Con(PA) 
> for S.
> Let's call that instance (neg Con(PA)-#). I make the following claim:
> (I) (neg Con(PA)-#) is true and assertable.
> Therefore
> CLAIM: (PA-#) is not a proper substitute for (#).
> This is my position. If you disagree with it, then you must either deny (I)
> or (II). I think that denying (I), or saying that it's not clear enough
> either to affirm or deny, is a truly desperate measure. If you deny (II),
>  In the end, none of
> the things I could think of allowed me to eliminate the counterfactual
> element. (This inability to find a suitable PA formalization of
> consistency-completeness was, by the way, a big part of my reason for
> wanting to discuss it on FOM.)
> If anyone disagrees with what I have been saying about
> consistency-completeness, I feel that they must be denying either (I) or
> (II). It seems that Joe Shipman and Torkel Franzen want to deny (I). Fine,
> if you can live with that, go ahead. > 

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