[FOM] a definable nonstandard model of the reals

[Philip Ehrlich]

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.