[FOM] Arithmetical soundness of ZFC (platonic)

[Nik Weaver]

I will reply to some of the comments that were posted in response
to my suggestion that ZFC may be consistent but not arithmetically
sound.  This and the following message will be my last posts on
the subject.

Thomas Forster points out that if ZFC is sound as a theory of
(well-founded) sets then arithmetical soundness follows.  That's
true, of course.  So if you believe that there is, in some sense,
a really existing canonical "world of sets" in which the ZFC
axioms hold, then you have no problem.  Well, you have a problem
with the classical paradoxes of naive set theory, but presumably
you're satisfied with the iterative conception or some other
account of sets that justifies ZFC but blocks the paradoxes.

But I wonder how you justify the power set axiom.  This strikes
me as a serious weakness in the iterative conception since it
seems to be simply taken for granted that we can "form" power
sets (whatever that means).  But I don't actually know of any
cogent justification of power sets that wouldn't apply equally
well to full comprehension.  When you put that together with
the fact that practically all mainstream mathematics can be
done without power sets, it really raises a question.  Panu
Raatikainen says that the power set axiom is "extremely natural".
I don't agree, but in any case isn't full comprehension just as
"natural"?  So that doesn't seem to be much evidence for soundness.

>From what I've read on the FOM list, I get the impression that
people basically fall into two camps.  Some want to carefully
build foundations up from the bottom, starting with principles
in which we have complete confidence and demanding a thorough
justification of any proposed extension.  I fall in this category.
So, for example, I accept countable constructions but I go to some
length to explain why (see Section 3.1 of "The concept of a set"
and Section 1.3 of "Axiomatizing ...").  I don't just say "it's
natural" or whatever and leave it at that.

The other group takes an "anything goes" attitude, the idea being
that we should look for the most powerful axioms we can find, and
if something turns out to be inconsistent we give it up.  That's
fine, but my feeling is that if that is your approach then don't
claim that what you're doing is arithmetically sound, because if
your axioms weren't sound you wouldn't have any reliable way of
knowing this.  As I pointed out in an earlier message, we expect
a generic formal system to be consistent but not arithmetically
sound, so a priori we expect "anything goes" people to end up
working with systems that are not arithmetically sound.

Nik

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
http://math.wustl.edu/~nweaver/conceptualism.html