[FOM] Completeness in non-standard analysis
Jorge M. Lopez
lopez.jorgem at gmail.com
Sun May 20 19:17:50 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 On page 47 of J.M. Henle and E.M. Kleinberg's Infinitesimal
Calculus the intermediate value theorem is stated and the proof is
presented in the pages that follow. The proof is a take-off from
Cauchy's old proof Cours d'Analyse Note III page 460. The statement:
Given a continuos real valued function f defined on the real interval
[a,b] such that f(a).f(b)<0, there is a real number c in the given
interval such that f(c)=0. 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. My question is
that it is not at all clear that the completeness of the real numbers
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? The above proof seems to
work fine if one begin with the ordered field Q and "constructs" the
corresponding hiperrational via the ultrafilter procedure. What is
going on? Thanks again.
Jorge M. Lopez
Departamento de Matematicas
UPRRP
Tel 787 281-0649
Fax 787 281-0651
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