[FOM] numbers and sets

[Hartley Slater]

Alan Hazen (FOM Digest Vol 9 Issue 22) seems to take Benacerraf's 
point about 'what numbers cannot be' to be merely that, while, say, 
von Neumann's chosen sets are not identical with the natural numbers, 
the latter are simply ambiguously any comparable series of sets.  He 
writes

>Benacerraf's problem, of the arbitrariness of the choice of sets to
>identify numbers with if arithmetic is to be "reduced" to set theory,
>though not generating much mathematical (or F.o.M.) interest, has been a
>thorn in the side of Platonistic philosophies of mathematics since he
>published "What numbers could not be" in 1965*.

No: numbers are not *any* series of sets, but instead the respective 
numbers of elements associated in various ways with certain series of 
sets.  On the Historia Matematica list at this very moment this issue 
is also being debated.  Robert Tragesser has gratifyingly recognised 
the great difference between numbers and sets, and John Mayberry has 
agreed with him, to a large extent.  Mayberry writes (HM V5 #95):

>2.) Frege's worry about the status of "abstract objects" like shapes
>and directions *as objects* bears directly on Robert Tragesser's
>comments on "abstraction" and "set theoretic reduction". It seems to me
>that his remark that "It is really logical sloppiness to
>set-theoretically represent a non-set entity by a set" fails to take
>into account the purpose of such representations. Naively the real
>numbers are not set-entities. But the reduction of real number theory
>to set theory does not consist in supposing, or stipulating, that they
>are.... Nobody would claim that real numbers
>are *really* Dedekind sections of rationals, or even that Dedekind
>sections of rationals can go proxy for (genuine) reals (so that we get
>a "reduction" of real number theory to set theory).

Mayberry, however, merely details the isomorphisms which commonly 
form the basis for identifications of numbers with sets, and does not 
spell out the limits of those parallelisms:

>The reduction employs the axiomatic method and can be divided into
>three steps: first, you determine the type of mathematical structure to
>which the theory applies (in this case the structure type of ordered
>fields) - this amounts to singling out the key notions of the theory
>(addition, for example). Second, you lay down axioms applying to
>structures of that type which correspond to true propositions in the
>naÔve theory; and you stop laying down axioms only when those you
>already have allow you to prove that all structures of the given type
>in which they hold true are isomorphic (i.e. you prove the axioms to be
>*categorical* - in the case of the real numbers you need the axioms for
>a complete ordered field). Third, and finally, you prove that the
>axioms are consistent by exhibiting a structure of the given type in
>which the axioms are satisfied. Here, and only here, do you use what
>Tragesser calls "set-entities" - Dedekind sections of rationals,
>equivalence classes of Cauchy sequences, finite von Neumann ordinals
>(if you are modelling the axioms of arithmetic), etc. But this is
>purely in the interests of producing a valid existence proof from
>purely set-theoretic assumptions.

All this is true, but still numbers are categorically different from 
sets, and just what their difference is needs to be made plain. 
Certainly there are similarities, but there is also a limit to those 
similarities.  The relation between a finite von Neumann ordinal, and 
the corresponding natural number, for instance, is that the ordinal 
contains that number of members: if the nth such ordinal is X then we 
can say '(nx)(x isin X)', using the exact numerical quantifier 
'(nx)'.  So the number corresponding to the ordinal Y is iota-n(nx)(x 
isin Y), and it is not just a 'naive' matter that iota-n(nx)(x is in 
Y) is distinct from Y.   That many people evidently have difficulty 
recognising the distinction also shows it has far more substance.
-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 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