[FOM] a definable nonstandard model of the reals
Philip Ehrlich
ehrlich at oak.cats.ohiou.edu
Tue Nov 18 15:14:54 EST 2003
John Baldwin wrote:
I don't find the solution by Kanovei and Shelah entirely
satsifactory for 2 quite different reasons.
1) The model is defined in set theory by essentially specifying how
to construct it. This is quite different
from defining the reals as the unique complete ordered field. To be
precise, is there a second order definition
in the language of fields which specifies a `canoncial' non-standard model?
2) It's the wrong model. At least they have only guaranteed an
omega-saturated model and I think we are looking for
at least an aleph_1 saturated one.
John,
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?
Best,
Philip Ehrlich
* "An Alternative Construction of Conway's Ordered Field No," Algebra
Universalis, 25 (1988), pp. 7-16. Errata, Ibid. 25, p. 233.
Also see my
"Absolutely Saturated Models," Fundamenta Mathematicae, 133 (1989), pp. 39-46.
"Universally Extending Arithmetic Continua," in Le Continu
Mathematique, Colloque de Cerisy, edited by H. Sinaceur and J.M.
Salanskis, Springer-Verlag France, Paris, 1992, pp. 168-178.
"Number Systems with Simplicity Hierarchies: A Generalization of
Conway's Theory of Surreal Numbers," The Journal of Symbolic Logic
66 (2001), pp. 1231-1258.
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