[FOM] real numbers

Arnon Avron aa at tau.ac.il
Sun May 25 07:29:32 EDT 2003

> Arnon Avron asks me (FOM Digest Vol 5 Issue 34):
> >In the first course on the calculus I learned in my very first week
> >as an undergrduate student in Tel-Aviv University the set of reals
> >was defined as the union of the set of the rational numbers and the set of
> >irrational Dedekind's cuts, which were introduced before (yes, at that time
> >in Israel students start learn Mathematics with Dedekind's cuts). The
> >definitions of <, + etc on R were given accordingly. The
> >idea, of course, was to have Q  as a subset of R (as all mathematicians
> >do) and still distinguish between the rational numbers and the
> >rational cuts (the definition of which assume the rationals). It
> >was noted also that for all practical purposes of Analysis
> >there is no real difference between these two structures.
> >
> >Would this procedure solve your problems?
> I'm glad at least someone can see one cannot define the rational cuts 
> until after one has defined the rationals.  Your old teacher's 
> definitions respected the obvious fact that one cannot have r={p|p<r} 
> without circularity, and so solved one problem.  

First I should clarify something. I have described how we were taught, but 
my teachers were hardly original here. They just followed one of the best
textbooks ever written: Fikhtengolt's classic two-volumes
(in English) "The Fundamentals of Mathematical Analysis" (translated 
in 1965 By I. Sneddon from Russian). This definition of the reals
is given in P. 5 of this book (which altogeter has about 1000 pages).
This book strictly distinguishes a rational number from the
corresponding cut.

> But there would 
> remain other problems, if the combined 'rational numbers plus 
> irrational Dedekind cuts' were taken to map onto the geometric line, 
> for instance.  Did your teacher also mark points on that line 
> supposedly corresponding to irrational cuts?  If so, did s/he also 
> draw your attention to the members of that point, and the members of 
> its complement?  Geometric points are not in the right category to 
> have members, or complements.  You might as well think that they had 
> negations, i.e. that '~r' made sense, where '~' is 'not'.

Here it is you who dont distinguish between numbers and points
It is generally taken for granted (not by me!) that
there  is a correspondence between the two (although
no *textbook* I know actually PROVE that this correspondence exists
and has the required properties), but they are not identical.
A number is a number (whatever this means). A point on
a line is a point on a line. In fact,
the book of Fikhtengolt's strictly follows the idea of "arithmetization
of mmathematics". Geometry is used in it only
as a tool for getting *intuition*, but
never as an official part of definitions or proofs (and this is 
how we were taught, and basically still teach ourselves).

Arnon Avron

More information about the FOM mailing list