[FOM] Tangential to Slater and Numbers

[Hartley Slater]

Allen Hazen opposes me (FOM Digest Vol 10 Issue 12):

>(In other words, I deny, what Slater has
>asserted in his latest (6.x.03) posting, that "with numbered
>variables there is a reference to numbers in the standard
>re-expression of numerical quantifiers.")

Perhaps Hazen also thinks that, in the diagonalisation of a list of 
functions fm(n) to obtain the function fn(n), the index 'm' likewise 
does not refer to numbers?  If that was so then, of course, the 
diagonal function of n could not be formed.  If one expresses 'there 
are exactly two Fs' as
(Ex1)(Ex2)(y)(Fy <-> y=x1 v y=x2)
then the counting of the variables is explicit, and so the number is 
referred to in the expression.

In any case I can make my main point - against von Neumann sets being 
numbers - quite independently of this, using Hazen's, or Holmes' 
offered logic in which the numeral in '(nx)Fx' can be quantified 
over.  For if at least this is so, then one can form 'iota-n(nx)(x 
isin {{}, {{}}})' in the Russellian manner, and how is this to be 
read?  It is to be read as 'the number n such that there are n 
members in {{}, {{}}}' i.e. 'the number of members of {{}, {{}}}'. 
But it is clear that, on either Hazen's or Holmes' construal, this 
number (which is the same as the number 2) is distinct from the set 
{{}, {{}}}, indeed from any set.
-- 
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