[FOM] Completeness in non-standard analysis

[Nigel Cutland]

The fact that there is a real number infinitely close to any finite
hyperreal (the Standard Part Theorem) is equivalent to the 
completeness of the reals, and this is what is needed to get c. 

_________________________________________

Professor Nigel Cutland
Mathematics Department
University of York
YO10 5DD
UK

e-mail: nc507 at york.ac.uk
Tel: +44-(0)-1904-433080
cell/mobile: +44-(0)7981-622828

-----Original Message-----
From: Jorge M. Lopez (by way of Martin Davis<martin at eipye.com>)
[mailto:lopez.jorgem at gmail.com] 
Sent: 21 May 2007 00:18
To: fom at cs.nyu.edu
Subject: [FOM] Completeness in non-standard analysis


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