[FOM] RE: Real Numbers
montez at fidnet.com
Tue May 6 10:53:25 EDT 2003
Debates over Benacerraf's problem always seem to lead to someone's
something like the following, which Wiman says here:
> 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.
I always feel puzzled when I get to that point, however.
So do I. But I have begun to formulate an idea I shall express below that
will hopefully clear this up. Let me point out that I can understand a
little bit of the above puzzling point in the following sense:
use the word ``the'' for whatever model they have in mind, as if everyone
is to understand that they are identifying all isomorphic copies of it.
may appear to some to be disingenuous when we are at other times being
for details, using only the article ``a'' when we know there is at least one
object with certain properties and do not (yet) know there is only one such
What is the
modality here? In what sense /can/ the reals be any of these things,
though they are not? I understand what it means to say, for example,
that, though I am not a mathematician, I might have been, but that is
surely not the sort of thing one has in mind here.
I would say you are correct. This is not the intended meaning in the above
puzzling statement. The fact that mathematicians use the phrase ``the real
numbers'' suggests that such mathematicians have in mind a particular
and that this particular structure is recognizable _by others_ in certain
specifiable structures such as the set of equivalence classes of cauchy
of rationals. Of course, in order to specify the structures _to others_ one
first be sure the others understand the foundations upon which one will
the representations. It is pointless to talk, to someone who knows nothing
the rational numbers, about cauchy sequences of rational numbers, much less
equivalence classes of such beasts. Of course, one person talking about
can never be absolutely certain the audience really already understands the
fundamental material for the discussion at hand.
If I am not a
mathematician, then perhaps I /could/ be one, but what would it mean to
say that I /can/ be one? Surely if the reals /are not/ Dedekind cuts,
then they /cannot/ be Dedekind cuts, any more than I can be George Bush
if I am not George Bush.
We can interpret analysis in set theory. That is true. But why should it
mean that the reals "can be" sets?
Now granted, there is a point at which one might be trying to get using
such language, and it is a point with real mathematical content. But I
do not see that it has been clearly expressed, either here or elsewhere.
Now, I shall try my hand at explaining what I think can clear this up.
My explanation will involve the use of urelements, which are objects
in a set theory that are not themselves sets. Such objects are at
times referred to as atoms. It has been understood for quite some
time that various set theories without urelements are as viable as
are corresponding set theories with urelements. For example, ZF and
ZFA (ZF with atoms) are, if I recall correctly, equiconsistent. Now,
as a mathemctician working with real numbers, I would not need to justify
to other mathematicians who work with real numbers my reference to
``the'' real numbers, for they are part of the given structure, which
can be understood as a particular structure given on a particular set
of urelements. Thus in a sense the reals are to such mathematicians
(i.e. real analysts) more fundamental than sets, and the set theory
ZFA can be augmented by axioms for making the reals be a specific
structure on (some of) the urelements. When being queried about
what are the real numbers, if the analyst wants to respond, s/he
may do so by producing models in ZF that are isomorphic to the
structure that is to the analyst more fundamental. The justification
for the analysts' use of the term ``the'' real numbers is that s/he
may conceivably add to ZFA a predicate Real(x) that specifies exactly
which urelements are real numbers, and then prove that the set of reals
in that interpretation is isomorphic (in every sense of the word) to
anyone else's chosen representation, in ZFA, which includes all the models
that may be constructed for the same structure under only the assumptions
of ZF. The analyst may not spend much time nowadays justifying the
use of the article ``the'' in reference to the object of discourse
of his/her profession. In fact, such justification is deemed unnecessary,
for one can construct a model, using dedekind cuts or equivalence classes
of cauchy sequences, and one can prove that the models are all isomorphic.
However, when the notions of dedekind cuts and cauchy sequences were
and models were constructed, it was perhaps unclear that the axioms analysts
were using were consistent, and the construction of models provided a
on that score.
However, we have some detractors that question the legitimacy of ZF who
would presumably also question the legitimacy of ZFA. Analysts then
will I think typically leave this debate, realizing there is no longer
anything of interest to them in it.
Associate Professor of Mathematics
Department of Mathematics and Statistics
University of Missouri - Rolla
Rolla MO 65409-0020
insall at umr.edu
montez at fidnet.com
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