# [FOM] Real Numbers

Lucas Wiman lrwiman at ilstu.edu
Fri May 2 17:40:45 EDT 2003

```In two recent postings, Hartley Slater accuses not only me, but a large
number of authors of set theory and real analysis textbooks of making a
systematic category error regarding the definition of the real number by
means of Cauchy sequences (or presumably Dedekind cuts, or any other set
theoretic account of the reals).  As I understand him, this is
essentially the same argument that Paul Benacerraf made in his famous
paper "What Numbers Could Not Be."  In this paper, Benacerraf argues
(among other things) that since any set theoretic representation of the
natural numbers includes some properties which numbers do not have (like
3 being an element of 5), numbers cannot be sets.  Slater argues along
similar lines that since equivalence classes of Cauchy sequences do not
sit in a line, and the real numbers do, that the reals cannot be classes
of Cauchy sequences.  Slater believes this because a class of Cauchy
sequences is not comparable with rational numbers, but only with an
equivalence class generated by what he calls a rational real (like 1/2,
1/2, 1/2,...).  A few points:

Why can't equivalence classes of Cauchy sequences sit in a line?  I can
certainly imagine identifying real numbers with sequences of rationals
converging to them (which do sit in a line), and hence why not look at
the real as being the same as the class of all of them?  There is a
quite attractive duality here.  We certainly do something like this in
projective geometry--one can think of a point (possibly ideal) as being
the same as all lines which contain it.  This leads to a beautiful
point-line duality, which allows us interchange point and line in
theorems of projective geometry.

When mathematicians (myself included) say equal or the same, we often
mean isomorphic.  Philosophers frequently miss this point.
Mathematicians don't care about these sorts of "category errors"
precisely because they are not errors.  Thus when people talk about
getting the same permutation group (on n elements) no matter what
n-membered set is used, they don't mean that all these groups are equal
in the philosophical sense.  Rather they mean that the group
theoretically relevant aspects of them are equal.  So when speaking
about group theory, why say they're different?  Much the same thing
happens here.  The differing axiomatizations of the reals become clearer
when viewed in this light.  An axiomatization for the reals in terms of
infinite strings of binary numbers shows the computational aspects of
the reals.  Cantor's axiomatization (above) shows the sequential
completeness, Dedekind's shows the least upper bound axiom.

Gian-Carlo Rota, David Sharp and Robert Sokolowski wrote an interesting
article on this topic (reprinted in Rota's book Indiscrete Thoughts).
Here is a quote from the introduction:  "The items of mathematics, such
as the real line, the triangle, sets, and the natural numbs, share the
property of retaining their identity while receiving axiomatic
presentations which may vary radically.  Mathematicians has axiomatized
the real line as a one-dimensional continuum, as a complete Archimedean
ordered field, as a real closed field, or as a system of binary decimals
on which arithmetical operations are performed in a certain way.  Each
of these axiomatizations is tacitly understood by mathematicians as an
axiomatization of the *same* real line.  That is, the mathematical item
thereby axiomatized is presumed to the *same* in each case, and an
identity is not questioned."  Near the end:  "Mathematicians are still
discovering new axiomatizations of the real line.  Such a wealth of
axiomatizations shows that to the mathematician there is only *one* real
line.  ...  The need for further axiomatizations is motivated by the
discovery of further properties of the real line.  The mathematician
wants to find out what *else* the real line can be.  He wants ever more
perspectives on the same real line."  Rota et al go on to define a
pre-axiomatic grasp of a subject, which allows the mathematician to see
mathematical objects defined in different ways as the same.  On most
points I agree with the conclusions of this article, and I highly
recommend it to anyone interested in the topic of mathematical identity.

So, yes, I agree with Slater.  The real numbers are not equivalence
classes of Cauchy sequences.  They can be, but they can also be Dedekind
cuts, binary numbers, real closed fields, one dimensional continua, or a
number of other things.

Regarding Slater's commentary on Dedekind, the creative power postulated
by Dedekind seems to be very similar to Rota et al's pre-axiomatic grasp
of a concept, though of course the language is a bit different.

- Lucas Wiman

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