[FOM] a definable nonstandard model of the reals

John T. Baldwin jbaldwin at uic.edu
Wed Nov 12 17:28:44 EST 2003

I thank Steve for pointing this interesting paper.

Stephen G Simpson wrote:

>Here is a paper that may be relevant to the earlier FOM discussion of
>nonstandard analysis.
> Title: A definable nonstandard model of the reals
> Authors: Vladimir Kanovei and Saharon Shelah
> Comments: (6 pages) to appear in JSL
> Subj-class: Logic
> MSC-class: 03H05
> \\
>   We prove in ZFC the existence of a definable, countably saturated elementary
> extension of the reals.
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.

Nor do I find the issue central to justifying nonstandard analysis.

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