[FOM] numbers and sets

Hartley Slater slaterbh at cyllene.uwa.edu.au
Sat Sep 27 23:22:47 EDT 2003

Allen Hazen asks (FOM Digest Vol 9 Issue 25):

>I would ask Slater to consider, however, whether the non-identity-standard
>interpretation of arithmetic in set theory (interpretation 4 in my post of
>a couple of days ago) doesn't go some distance toward meeting the objection.

Of course it goes a long way.  And I also agree with

>  A natural way to formalize the
>three-way distinction of true/false/categoreally-odd is to use a
>MANY-SORTED logic, with notationally distinguished alphabets of variables
>for numbers and for sets, and with predicates that can only form
>well-formed atomic formulas with terms of the appropriate sort. (Category
>mistakes, in such a formalization, become syntax errors.)

But I also think there is no choice about the matter: the place for 
'n' in '(nx)(x is in Y)' has a quite different syntax from the place 
for 'Y', and any formalisation which does not respect this difference 
is not going to mix up numbers and sets, since, for instance, 
iota-n(nx)(x isin {{},{{}}}) is identical to the number 2, but is not 
identical to {{}, {{}}})

I have dealt with Randall Holmes main point before.  He presents an 
argument including:

>Premise 1:  The natural numbers are (individually) properties of finite sets.
>Premise 2: (Selected) properties are to be identified with their
>extensions (with the set of objects having that property).
>Premise 3:  the finite cardinalities are properties of the kind
>whose extensions are sets

But premise 2 is false, since, following on from the above, the 
number two is not the set of things with two members, and so it is 
not the extension of the property of having two members - it is the 
extension of the property of being 2!
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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