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


MoeBlee wrote:
> Rupert wrote:
> > > So any model of those axioms is a real closed field but not every model
> > > of those axioms is a complete ordered field. Do I have that right?
> >
> > Yes.
> >
> > > So
> > > it's a twist, no?, that it turns out that if you take the definition in
> > > set theory of 'a complete ordered field' and restate it as first order
> > > axioms, then its class of models is the class of real closed fields
> > > instead of the class of complete ordered fields.
> > >
> >
> > Well, it's not that surprising. The first-order axiom schema states
> > that the least upper bound principle applies to those sets which are
> > parametrically definable by formulas. There are only countable many
> > such sets. So it's not surprising that this should turn out to be
> > weaker than the full second-order least upper bound principle.
>
> Right. But does that mean that all real closed fields are complete
> ordered fields, but not vice versa? I'll address these fields in my
> studies later. But for the purpose of this discussion about
> isomorphisms and models, this would help to know.
>

All complete ordered fields are real-closed fields (in fact, all
complete ordered fields are isomorphic to R), but not all real-closed
fields are complete ordered fields.

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