[FOM] predicative foundations

[Stephen Pollard]

On Wed, 15 Feb 2006 Nik Weaver wrote:

>You can't even say what P(N) is without resorting to platonistic ideas.  I
>can describe N in a metaphysically uncontroversial way in terms of making
>marks on paper.  But there is no, or at least no obvious, way to say what
>P(N) is without invoking "sets of numbers" as abstract entities.

Here is one way of characterizing the sets of natural numbers. Bear with me
while I make some juvenile-sounding observations. I'm just trying to make
sure I'm understood.

Note, first, that some properties are non-distributive: a non-distributive
property can apply to several things without applying to any one of those
things. Jane can be one of the ten police officers who surrounded the
house, even though Jane did not herself surround the house.

We can understand "form the set S" to be a non-distributive property of S's
members. The prime numbers form the set of primes, but no one prime number
does so. 17 does not form the set of primes because not every member of
that set is 17. Though it may sound a bit odd, 17 does form the set {17};
"form" does not mean "is identical with."

Say that some numbers form a set if and only if there is a set that they
form. We can now say what sets of numbers there are by saying which numbers
(collectively) form sets. The classical answer is: any numbers will form a
set.

Let epsilon be the relation that holds between a number n and a set of
numbers S when n is one of the numbers that form S. Assume an
extensionality axiom.  I have now characterized the structure
<N-union-P(N),epsilon> without, as far as I can tell, resorting to
platonistic ideas. One may need to resort to such ideas to answer certain
philosophical questions. My very limited point is that one can describe the
structure at issue without first settling the metaphysical controversies.
Deciding whether to believe that the structure exists or has instances is
another matter.

Weyl would insist that I don't really understand the assertion, "Any
numbers will form a set." I'm pretty sure he would not say that settling a
metaphysical controversy would help to improve my understanding.

Stephen Pollard
Professor of Philosophy

Division of Social Science
Truman State University
Kirksville, MO 63501

(660) 785-4653