[FOM] Slater and numbers

Hartley Slater slaterbh at cyllene.uwa.edu.au
Thu Oct 2 20:20:02 EDT 2003

Randall Holmes now writes (FOM Digest Vol 10 Issue 4)

>1.  According to standard logical usage, what Slater writes
>(nx)(Px) does _not_ contain a quantifiable place "n".
>2.  Leaving the Frege numerals entirely aside, we discuss how
>things are done in ZFC.  Define 0 as {} (the empty set) and
>define x+ as x \cup {x} as usual.  The Axiom of Infinity combined
>with Separation tells us that there is a unique set \omega = N
>which is the intersection of all sets which contain 0 and are
>closed under the successor operation.  Elements n of this set
>N are called "natural numbers".  To say that a set A has n members
>is to say that there is a bijection between A and n.

On 1, see my previous remarks about appeals to what is standard, 
also, for instance, Allen Hazen's recent contribution (FOM Digest Vol 
10 Issue 3) about quantifying over numerical quantifiers (and the 
details in Bostock's Ch 3 previously referred to).  On 2, I dealt 
with this before: the empty set is merely a canonical set with zero 
members and so is not the number itself.  If it were zero itself then 
'To say that a set A has n members is to say that there is a 
bijection between A and n' would have the consequence, for instance, 
'To say that a set A has {} members is to say there is a bijection 
between A and 0'.  There is no problem, of course, with such things 
as 'To say a set has 2 members is to say that there is a bijection 
between A and {{}, {{}}}'.  The problem is with taking the set to be 
the number.  So when Holmes goes on:

>Most fundamentally, and crucially for Slater to understand why his
>arguments are unconvincing, the standard definitions of the natural
>numbers have nothing to do with numerical quantifiers.  One cannot
>argue from the grammar of numerical quantifiers to any position about
>what numbers can or cannot be unless the point is granted that the
>numbers must be defined in terms of numerical quantifiers, and it
>isn't -- not by me, and in fact not by anyone to speak of in
>mathematics or logic.

he is presenting no argument against me, but merely re-iterating the 
opinion I have argued against.
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

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