# [FOM] what numbers can be

Hartley Slater slaterbh at cyllene.uwa.edu.au
Wed Sep 24 02:33:51 EDT 2003

```At 12:02 AM -0400 24/9/03, Randall Holmes wrote:

>There is a set theoretical definition of the natural numbers which is
>arguably not artificial at all.  This is Frege's definition which has
>the effect of defining each concrete natural number n as the set of
>all sets with n elements. this is of course not the formal definition
>-- it would be circular -- but a formal definition with this effect is
>straightforward).

>To understand "the set A has three elements" as meaning "A belongs to
>3" is much more natural than to understand it as meaning "there is a
>bijection between A and {0,1,2}".

The problem is not with defining 'the set A has three elements' as
'The set A is a member of the set of those sets with three elements'.
The problem is with saying the latter set is the number three.  And
the problem with that is not the circularity, but with the grammar of
'the number three'.  3 squared, for instance is 9, but you cannot
square a set - nor can you take its square root.

>Conclusion:  there is no knockdown argument for Slater's assertion that
>numbers are not sets.

The point about squares and square roots is one such; I presented
several other similar category mistakes in earlier postings.
--
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

```