[FOM] Could PA be inconsistent?
Richard Heck
heck at fas.harvard.edu
Fri May 14 13:03:04 EDT 2004
> There are now two threads going on FOM contemplating this possibility.
> No one has mentioned that we possess proofs that PA is consistent.
> These proofs can be carried out in quantifier-free primitive recursive
> arithmetic with induction permitted on a primitive recursive
> well-ordering of the natural numbers of order type epsilon-0. These
> proofs (Gentzen, Ackermann, G\"odel) have gone unchallenged for decades.
>
> Martin
Crispin Wright once compared the situation here to that with
philosophical skepticism. We have what we take to be good reasons to
believe e.g. that the speed of light is finite, that the sun will rise
tomorrow, and, indeed, that there are other people in the world. But, as
Descartes famously pointed out, our reasons could conceivably be
revealed as fraudulent. It has now become the common wisdom in
epistemology, however, that the mere fact that such an eventuality is in
some sense conceivable does not undermine our claim to know such things
as those mentioned. For similar reasons, the mere fact that the proofs
of PA's consistency Prof Davis mentions depend upon assumptions that
are, in a reasonably well-defined sense, stronger than those of PA
itself does not undermine the reason-giving force of the proof. /If/ PA
were shown to be inconsistent, then the reasons we take ourselves to
have for believing it is consistent would, of course, thereby have been
shown to be fraudulent. But that is no more interesting a fact than
that, /if/ it were shown that all my experience was the result of an
evil demon's manipulations, my reasons for believing Prof Davis exists
will have been shown fraudulent.
Richard Heck
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