[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
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