``PA is consistent'' - Was Re: [FOM] Proof "from the book"

[Michael Kremer]

Insall's question was undoubtedly confused.  His post read:

"Are you saying that PA is (definitely) consistent? If not, then I 
misunderstand you, for it seems to me that only consistent theories have 
the property you claim for PA above: ``proves only true sentences''. 
(Exercise: Can a consistent theory prove a false sentence?) If you are 
claiming that PA is (known to be) consistent, then how does your proof of 
the consistency of PA go? In what theory does your proof reside?"

In context, the only "exercise" that makes sense would be "can an 
inconsistent theory prove only true sentences" since a consistent theory 
proving a false sentence would not in any undermine the claim that only 
consistent theories prove only true sentences (though it would undermine 
the claim that all consistent theories prove only true sentences).

But you don't have to go to theories like PA + PA is consistent or the 
equivalent for Robinson's arithmetic, to show that a consistent theory can 
prove a false sentence.  This fact should be accessible to elementary logic 
students with almost no knowledge of mathematics at all.  For example, the 
theory
         {Al Gore is President}
is consistent and proves the false sentence
         Al Gore is President (formalized as "Pa").

Or, if you prefer, the theory
         {1<0}
is consistent, and proves the false sentence
         1<0.

--Michael Kremer.


At 02:03 PM 9/2/04 -0700, you wrote:



>On Thu, 2 Sep 2004, Matt Insall wrote:
>
> > (Exercise:  Can a consistent theory prove a false sentence?)  If you are
> > claiming that PA is (known to be) consistent, then how does your proof of
> > the consistency of PA go?  In what theory does your proof reside?
>
>         The theory PA + (not "PA is consistent") is consistent if PA is
>and proves the false stament "PA is not consistent".
>
>         Since the poser of this question views the consistency of PA as in
>doubt he may find this example unconvincing. One can replace PA by
>Robinson's system Q in the above example. The resulting argument has a
>proof formalizable in "primitive recursive arithmetic". {Generally
>considered the touchstone for "finitist proofs".}
>
>         --Bob Solovay
>
>
>_______________________________________________
>FOM mailing list
>FOM at cs.nyu.edu
>http://www.cs.nyu.edu/mailman/listinfo/fom
-------------- next part --------------
An HTML attachment was scrubbed...
URL: /pipermail/fom/attachments/20040903/3cc4de21/attachment.html