[FOM] Completeness in non-standard analysis

Robert Lubarsky robert.lubarsky at comcast.net
Mon May 21 07:24:14 EDT 2007

```I'm not familiar with the Henle-Kleinberg book, but can say some things in
general anyway about the given proof of the Intermediate Value Theorem.

> The procedure for the proof divides the
> interval in n equal parts (n a positive integer) and argues that f
> must change parity over one of the subintervals. This statement
> remains true for the hiperreals and it must be true for an infinite
> hipernatural number N. Then in one of the resulting subintervals (of
> infinitesimal length) there must be a real number and it is fairly
> easy to see that this number is the desired number.

Do you mean there must be a standard real number in each interval? Not so.
Let c>0 be infinitesimal. Then (c/2, c) has no standard reals. Rather, take
the standard part of the hyperreals in any such interval. Since the interval
has infinitesimal length, there is only one such standard real.

> My question is
> that it is not at all clear that the completeness of the real numbers
> gets used at all.

That the standard part is a real number. In contrast, for example, with the
rationals. Some hyperlong rational approximation to the square root of two
would not have a rational standard part.

> Some related questions are as folows: What is it
> known of the cardinality of the ultrafilter used in the development
> of the hiperreals?

Anything you want.

> Are the hipernatural numbers and the hiperintegers
> sets with the cardinality of the continuum?

I consider the use of ultrafilters to be a bit of a red herring. By the
Upward Löwenheim-Skolem Theorem, you can get such structures of any size you
want.

Bob Lubarsky

```