[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
theory.
> 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
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