[FOM] Real numbers

[Hartley Slater]

John Pais (FOM Digest Vol 5 Issue 12) happily finds no category 
mistakes in real number theory, as it is developed in Walter Rudin's 
standard text.  But that is because Rudin is careful not to identify 
what is merely isomorphic, and my original claim was merely that 
there would be a category mistake in doing the reverse.  So there 
still is a need to take care.  For instance Rudin says

>p. 20:

>"Step 8 We associate with each r in Q the set r* which  consists of 
>all p in Q such that p < r...."

>p. 21:

>Step 9  We saw in Step 8 that the replacement of the rational numbers [each
>rational number] r by the corresponding 'rational cuts' r* in R 
>preserves sums,
>products, and order.  This fact may be expressed by saying that the ordered
>field Q is *isomorphic* to the ordered field Q* whose elements are rational
>cuts. Of course, r* is by no means the same as r, but the properties we are
>concerned with (arithmetic and order) are the same in the two fields.

Indeed, the cut {p|p<r} has a complement {p|~(p>r)}, while the 
rational number r not only hasn't a complement, but is not even the 
sort of thing which could have a complement, so they are in different 
categories, and it would be a mistake to identify them.  Likewise 
with the rational number r and the equivalence class associated with 
it, namely [<r,r,r,...>], i.e. the class of all Cauchy sequences 
equivalent to the sequence consisting in the given rational number 
endlessly repeated.  One categorical difference here is that the 
equivalence class has members while the rational number does not. 
There is nothing on the geometric line which corresponds to the 
inside of an equivalence class, even if the appropriate equivalence 
classes map onto all the points on that line.  Cauchy, for instance, 
took infinitesimals to be members of [<0,0,0,...>], but they have no 
decimal representation.  The rational number zero has no inside.
-- 
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