[FOM] Real Numbers

[Hartley Slater]

Lucas Wiman (FOM Digest Vol 5 Issue 4) has got me wrong, in a number 
of ways.  He says

>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.
>....... 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.

No, my point does not resemble Benacerraf's.  I was not saying that, 
since there are varying definitions of the reals, no one definition 
can claim any special precedence.  In my posting about the Cauchy 
definition (FOM Digest Vol 4 Issue 35) I was just speaking about the 
Cauchy definition (see again 'Axiomatic Set Theory' by Suppes, 
specifically p181).  And I have no objection to that definition other 
than the fact that it *alone* does not produce items which are 
comparable to rationals.  I argued that a category mistake is 
involved in thinking classes of Cauchy sequences *themselves* are 
comparable to the rationals.  Specifically, it is the rational reals 
which can be less than, equal to or greater than other reals, and a 
rational real is not in the same category as a rational  - so one 
cannot, for instance, have,: '2=[<2,2,2,...>]'  (in Suppes' 
symbolism, '<...>' forms a sequence, and '[...]' the associated 
equivalence class).  If further argument about the impossibility of 
this identity is required, consider that unless the rational number 2 
was defined differently (as indeed it is in Suppes, see p110), one 
could not go on to define the infinite sequence within the rational 
real in terms of it, as in Suppes' definition 56, p182.

>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.

No, it's not a matter of some pre-axiomatic grasp of a concept.  The 
Dedekind material from Grattan-Guinness, in FOM Digest Vol 5, Issue 
2, was useful because it paralleled my point, using a different 
definition of the reals.  Dedekind appreciated that to get something 
comparable to the rationals, in connection with his cuts, you could 
not *just* use the cuts.  You had to have something 'corresponding 
to' the cuts as well; since even a rational number was 'not identical 
with the cut generated by it'.  The full quotation showed people 
around his time did not like his assumption of some 'creative power' 
to get these other entities, and indeed there is one place where he 
backed down on that, as Grattan-Guinness records (p224 again):

>in a letter to Lipschitz [Dedekind] remarked parenthetically that if 
>one does not wish to introduce numbers in his sense, "I have nothing 
>against it; the theorem I prove [on completeness] then reads: the 
>system of all cuts in the discontinuous domain of rational numbers 
>forms a continuous manifold"

Russell's identification of cuts with numbers, and Dedekind's 
proposed creation of 'corresponding' numbers, would both have enabled 
the theorem to be upgraded to "the system of all numbers which 
consist in, or correspond to, cuts in the discontinuous domain of 
rational numbers forms a continuous manifold".  But Dedekind backed 
down to the safety of just saying something about cuts (and so, 
apparently, he was not committed to an extension of the number 
system).

To make my point again, by means of another example:
No equivalence class is comparable to any rational number, since they 
are in different categories.  Suppes, for one, makes an associated 
category mistake when giving a proof which tries to show that, for 
instance, every real number is representable as a decimal.   Maybe it 
is so *representable*, but his generalised proof, for any integer 
radix r greater than 1, assumes, for a start (p190), that, given a 
real number x, there is an integer a such that ar is equal to or less 
than xr, making a the largest integer equal to or less than x.  Here 
'x' cannot be an equivalence class of Cauchy sequences of rationals, 
i.e. something like '[<...>]', otherwise it would not be in the right 
category.  So the proof Suppes provides is faulty.

Sandy Lemberg writes (in this issue?):

>Your view of Q<R as fundamentally different from Q in and of itself, and
>that identification of the two Q's is a "category mistake", misses the
>point. It is all pervasive in mathematics to make such identifications,
>again because it is the structure and not any characteristics of its
>elements or "points" which is of interest. Thus, such identifications are
>not "category mistakes" at all, but rather commonplace mathematical
>practice.

So cuts are equivalence classes?  It would be a mistake to identify 
them.  Following on from the above, there are three relevant 
structures, one containing equivalence classes, one containing cuts, 
and one containing rational numbers.  Certainly the first two are 
isomorphic.  Dedekind's point was that the latter two were not 
provably isomorphic without a further assumption, since one needs 
elements in the third corresponding to irrational cuts.  The category 
mistake he pointed out consisted in identifying elements in the 
second structure with elements in the third.  The category mistake I 
pointed out repeated his in the case of the first and third 
structures.
-- 
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