[FOM] real numbers

[Arnon Avron]

Hartley Slater write:

> The points I have made were that *the natural numbers* and *the 
> rational numbers* were not Dedekind cuts, i.e. that 'r={p|p<r}' is 
> false - that would make those numbers objects of a certain sort.  And 
> the 'core mathematician' Suppes was quite clear that rational numbers 
> are not the same as rational real numbers, as I also pointed out.  I 
> am sure, nevertheless, that Steiner is right about the sociological 
> fact that many would resolutely find no problem with 'r={p{p<r}'.  No 
> one likes to admit they have made a mistake, especially not as big a 
> mistake as Frege made.

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?


Arnon Avron
School of Computer Science
Tel-Aviv University