[FOM] What does Peano arithmetic have to offer?

[Monroe Eskew]

On Fri, Apr 30, 2010 at 9:21 PM, Vaughan Pratt <pratt at cs.stanford.edu> wrote:
> 3.  While FOM has its misgivings about category theory as an alternative
> to elementary theories of mathematical domains, it seems to me that
> essentially the same misgivings ought to be triggered already by
> abstract algebra, which predates CT by quite a few decades.  (Not to be
> confused with elementary algebra, which is just quantifier-free
> equational logic, well inside the limits of first-order logic.)  The
> homomorphisms, quotients, products, tensor products, function spaces,
> etc. of abstract algebra fall well outside the scope of first order
> logic.  Yet even taken together they cannot be said to be anywhere near
> the power of full-blown second order logic, even though the instincts of
> a first order logician would be to classify abstract algebra as a second
> order system on the ground that it is not first order.

I don't understand; abstract algebra can be done within set theory
which is a first order theory.

Best,
Monroe