[FOM] What does Peano arithmetic have to offer?

[Monroe Eskew]

On Mon, May 3, 2010 at 10:45 AM, Vaughan Pratt <pratt at cs.stanford.edu> wrote:
>
> What I don't see is whether Harvey's theorems 4 and 5 are sufficiently
> robust as to carry over to algebraic number theory in any form a number
> theorist would understand.  If not then they would seem to be the sort of
> theorems that would appeal only to logicians and those number theorists
> still working inside PA.  Algebraic number theorists consider that they're
> doing number theory, which raises the question, what can logicians tell
> number theorists they can't do?

I think what they can tell them is this:  You can't solve all
number-theoretical problems unless you make increasingly stronger
assumptions.  They might also be able to say things like, "That strong
assumption was not necessary; we can prove it in this weaker theory."
I think PA is relevant because it is a relatively weak theory but a
necessary starting point for number theory (in that number theorists
will accept PA as true of N), and there is no way to leave that
starting point without bringing Goedel's incompleteness theorems along
with you.