[FOM] real numbers

[Hartley Slater]

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.  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'.
-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html