FOM: Existence of Axioms and Existence of Collections of Axioms

Matt Insall montez at rollanet.org
Tue Nov 7 08:59:40 EST 2000

A question that comes to my mind lately, in a foundational sense, is the
following:

If one is ontologically committed to the existence of the theory PA, is one
necessarily ontologically committed to the existence of an infinite set?

As a consequence of considering this question, I am lead to the following
question:

If one is ontologically committed to the finiteness of all sets, is one
necessarily ontologically committed to the finite axiomatizability of each
theory to whose existence one has an ontological committment?

Similarly, let S be the set theory whose axioms are those of ZF, but not
including the axiom of infinity.  As I understand it, S has infinitely many
axioms.  Now if one is committed to the existence of the collection of one's
axioms, and one is committed to S, then does this not entail an ontological
committment to the consistency of ZF?  It seems to me that one can have a
model for \omega built from the symbols and formulas of S.

Matt Insall