Re: What is the 1st order formal system known as PA?


MoeBlee wrote:
> Yes, I overlooked that. To speak of bounds on subsets of the domain, we
> have to speak of sets, so that we're in set theory or we can cast the
> matter in second-order. But, to express subsets of the domain couldn't
> we use an axiom schema with phi, for formulas, instead of using sets or
> quantifying over second-order predicate symbols; thus we could express
> the axioms of a complete ordered field as a first order theory with
> axioms and one axiom schema?
>

I think, in fact, this theory you're talking about is the same as the
theory of real-closed fields. These are the ordered fields in which
every positive element has a square root and every polynomial of odd
degree has a root. The theory of real-closed fields is actually
complete. There is no contradiction with Goedel because the notion "is
a natural number" cannot be defined in the first-order language of the
theory of real-closed fields, so arithmetic cannot be embedded in the
theory. And it is rather trivial to prove there exist nonisomorphic
real-closed fields.

.