[FOM] links between nonstandard analysis and formal logic and settheory

Robert Lubarsky robert.lubarsky at comcast.net
Tue Jan 10 08:39:47 EST 2006

Ben Crowell asked about "the relationship between the notion of an
internal set in nonstandard analysis and internal set theory in general."

Suppose you have a non-standard, non-omega-standard extension of the
universe of sets V. (For instance, take an ultrapower of V by some
non-principle ultrafilter on omega.) Then the reals in that extension are a
non-standard model of analysis. (Since the omega of the extension is
non-standard, let n be a non-standard integer; 1/n is then an
infinitesimal.) In this model, any notion in non-standard analysis is the
restriction to analysis of the corresponding notion in non-standard set

> I'm also interested in trying to absorb the paper by Kanovei
> and Shelah, "A definable nonstandard model of the reals,"
> http://shelah.logic.at/files/825.pdf , but it seems like I need to bone
> up on my formal logic and set theory, since there are a lot of terms
> on the first page of the paper that I don't know. Can anyone recommend
> a good book that a nonspecialist could read for background?

A good reference for the model theory in the paper (e.g. saturation and
elementary extensions) is Chang & Keisler, as listed in the paper's
references. For the set theory (canonical well-ordering of L,
inaccessibility) a standard reference is Jech, although that might not be
the best for background for a non-specialist. Easiest is to ask somebody
nearby to sit down with you for an hour and explain what you need.

Bob Lubarsky
Florida Atlantic University

More information about the FOM mailing list