FOM: Dedekind
Kanovei
kanovei at wminf2.math.uni-wuppertal.de
Thu Mar 12 12:41:25 EST 1998
>Date: Wed, 11 Mar 1998 12:44:13 -0800
>From: Vaughan Pratt <pratt at cs.stanford.edu>
>The fact that the complement of the cuts would work just as well as the
>cuts themselves as representations of the reals should make it clear
>that it is meaningless to identify the reals with the cuts.
This everything is meaningless because the
"complements of cuts" is a meaningless notion.
Indeed, a CUT (or: Dedekind cut) is a partition
of the rationals onto two disjoint sets A and B,
such and such ... .
Dedekind was very clever. He knew that prominent
professors would ask why he had taken A rather than B
to represent a real, or vice versa. So he took the
pair of A and B, leaving for the nowadays categorists
to be kidding over their own interpretations.
Now come back to Dedekind. I take the following from
a letter by W. Tait.
>Dedekind's answer is that they [real numbers]
>are simply the numbers. The proper answer to *what* they are is
>answered---as he answered it---by exhibiting the structure of
>the system of numbers.
This is a good practical point of a real analyst:
there is a system of real numbers, which one can explore.
In particular one of Dedekind's own contributions in
this study was that reals are in a 1-1 correspondence with
cuts of rationals, just similarly .
This works perfect until one comes to a more phylosophical
point: is there something like a common background that
can gather different branches of mathematics like e.g.
real analysis and geometry. (Clearly not many of
mathematicians come to this point, of course.)
Now, a set theorist comes and says that such a common
background exists, and it is enough to get an
adequate model of any existing mathematical structure,
in particular, of the reals. This model is
<R ; <,+,x, ...> (*)
where R is the set of all Dedekind cuts in Q while
< etc. are properly defined such and such, and after
all this is so a comprehensive model for the system of real
numbers that one loses merely nothing if saying that
the reals ARE the cuts BY DEFINITION, just for the
sake of simplicity. But saying that the cuts MODEL the
reals would be more philosophically sound, I think.
In fact this consideration is similar to the following:
is a rational a fraction of two integers or such
fractions only model rationals. Why nobody of categorists
is playing this deepest of problems ?
The question whether e is a subset of \pi is meaningless:
to be a subset is not a property meaningful for (*).
This is meaningless, in the same sense as if one
would put e and \pi in incomparable types of reals just
because "e" is Roman while "\pi" is Greek.
Of course one gets many isomorphic but set
theoretically different models for the reals.
(Even more, different constructivists and categorists add a
variety of non-isomorphic ones, but this is another story.)
If this causes some philosophical trouble, just take the
isomorphism class, that is a well known approach since
long ago.
Vladimir Kanovei
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