[FOM] predicative foundations

[Stephen Pollard]

On Fri, 17 Feb 2006 Nik Weaver wrote:

>Stephen Pollard suggests a possible way to get around my claim that "You
>can't even say what P(N) is without resorting to platonistic ideas." I
>want to understand this better.  It seems like the proposal involves the
>concept "set" as an undefined primitive concept. Is that a fair assessment?

That does sound fair. Consider the following two approaches to the
foundations of set theory.

(1) We might treat the expression "set of numbers" as sufficiently well
understood and freely use it when we characterize the structure-type
<N-union-P(N),epsilon>.
(2) We might characterize the structure-type <N-union-P(N),epsilon> without
invoking "sets of numbers." (My earlier post sketched out one way of doing
so.) We could then introduce the expression "set of numbers" as a technical
term that helps us talk about <N-union-P(N),epsilon>. (In any structure of
this type, some things stand in the relation epsilon to other things. The
latter things are the "sets of numbers" of that structure.)

Strategy 2 is known in the trade as a "structuralist approach to set
theory." Such "structuralisms" have been hot topics in the philosophy of
mathematics for several decades.

Suppose we have done everything described in approach 2. We could still do
plenty of philosophical work. Here are some questions we might pursue. Are
there any structures of type <N-union-P(N),epsilon>? Could there be? If
there could, would they occupy possible worlds we can clearly conceive? If
our concept of <N-union-P(N),epsilon> is not hopelessly muddled, for what
purposes is it sufficiently clear? Clear enough for philosophy? For
mathematics?

It would not be crazy for someone to insist that he can clearly conceive of
possible worlds occupied by structures of type omega, but cannot clearly
conceive of possible worlds occupied by structures of type
<N-union-P(N),epsilon>. Prof. Weaver: Are you such a person? Am I getting
close to expressing your position or am I missing the point?

Stephen Pollard
Professor of Philosophy

Division of Social Science
Truman State University
Kirksville, MO 63501

(660) 785-4653