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

Michael Kremer kremer at uchicago.edu
Fri Sep 3 08:54:38 EDT 2004

```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
>
>
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