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

Robert M. Solovay solovay at math.berkeley.edu
Thu Sep 2 17:03:42 EDT 2004

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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