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

Aatu Koskensilta aatu.koskensilta at xortec.fi
Fri Sep 3 08:25:31 EDT 2004

Matt Insall wrote:

>If you are
>claiming that PA is (known to be) consistent, then how does your proof of
>the consistency of PA go?  
The axioms of PA are true in the natural numbers. Rules of inference of 
first order logic
preserve truth and no conradiction is true, hence PA does not prove a 
contradiction. You can of course
doubt the soundness of this argument, but if you do, then it seems you 
have much more important
things to complain about, e.g. well-defidedness of the concept of 
natural number or something like
that. Accepting ordinary mathematical reasoning about natural numbers 
while claiming that there's
something fishy about the consistency of PA is disingenious.

>In what theory does your proof reside?
In ordinary mathematics.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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