[FOM] Completeness in non-standard analysis (Jorge M. Lopez)

David Ross ross at math.hawaii.edu
Mon May 21 04:41:54 EDT 2007

> In reading some of the non-standard proofs of the intermediate value
> theorem, it is not clear at all how the hypothesis of completeness
> gets used

Completeness shows up when you take the standard part map and get a real
number.  If X is an incomplete metric space, say <x_n> a standard Cauchy
sequence with no limit in X, then for H an infinite natural number the
element x_H of *X will not have a standard part.

> gets used at all. Some related questions are as folows: What is it
> known of the cardinality of the ultrafilter used in the development
> of the hiperreals? Are the hipernatural numbers and the hiperintegers
> sets with the cardinality of the continuum?

There is no single nonstandard model; by appropriate choice of
ultrafilters/construction one can give the hyperintegers arbitrarily
large external cardinality.

David R.

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