[FOM] re: a definable nonstandard model of the reals
Dave Marker
marker at math.uic.edu
Wed Nov 19 14:34:41 EST 2003
Philip Ehrlich wrote:
>Since Conway's ordered field No is a nonstandard model of the reals,
>doesn't the following characterization of No I published back in
>1988* provide a positive answer to your recent question about a
>definable nonstandard model of the reals?
>No is (up to isomorphism) the unique real-closed field that is an
>eta_On ordering, i.e., No is (up to isomorphism) the unique
>real-closed field such that for all subsets L and R of the field
>where every member of L precedes every member of R , there is an
>element y of the field lying strictly between those of L and those of
>R.
>Of course, No is a proper class, not a set. Is that a problem for you?
This gives some kind of cannonical real closed field, but, as I understand
it, the problem is to look for a nonstandard model of the reals where we
have predicates for all subsets of R^n.
If we are only looking at the ordered field structure, there are many
natural nonstandard models. For example, the real closure of R(t) where
t is a positive infinite element. This could also be described
algebraically as the the field of algebraic Puiseux series (or we
could look at the larger models of formal or convergent Puiseux series).
While there is a notion of "integer" in these models, it is very poorly
behaved (unless you want the square root of 2 to be rational).
Dave Marker
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