[FOM] a definable nonstandard model of the reals

[Stephen G Simpson]

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.
   It seems that it has been taken for granted that there is no distinguished,
 definable nonstandard model of the reals. (This means a countably saturated
 elementary extension of the reals.) Of course if V=L then there is such an
 extension (just take the first one in the sense of the canonical well-ordering
 of L), but we mean the existence provably in ZFC. There were good reasons for
 this: without Choice we cannot prove the existence of any elementary extension
 of the reals containing an infinitely large integer. Still there is one.
   Theorem (ZFC). There exists a definable, countably saturated extension R* of
 the reals R, elementary in the sense of the language containing a symbol for
 every finitary relation on R.
 \\ ( http://arXiv.org/abs/math/0311165 ,  9kb)