[FOM] paper announcements

Nik Weaver nweaver at math.wustl.edu
Wed May 20 04:28:31 EDT 2009

In response to message # 013641 from Tim Chow:

Tim, thank you for your comment.  You say that you have
trouble following my reasoning for doubting that ZFC has
a model with a standard omega.  Even granting that we have
no special reason to expect such a model exists, you ask
"why shouldn't we simply be agnostic about the existence
of a model with standard naturals?"

Well, because that's a very special property.  Whatever
r.e. first order axioms you take for the natural numbers,
there are going to be lots of nonstandard models but
only one standard model.

It's not so surprising for a formal system to be
consistent.  All you need is that each axiom not be
the negation of a theorem provable from earlier axioms.
To be sound with respect to the target structure that
you're modelling is a much stronger condition because
each axiom has to actually be true.

E.g., if you start with Peano arithmetic and randomly
add ten new formulas as axioms, the chance that what
you get is sound is only 2^{-10} but the chance that
what you get is consistent is close to 1.  Or maybe it's
exactly 1, depending on how we define "random".  There
should be a way of making this assertion rigorous.

In the case of ZFC, if you're not a platonist then
there is no target structure.  There's just a list
of axioms which may well be consistent but which we
have no special reason to believe are arithmetically

You were especially interested in one comment I made
about recursive complexity.  You wrote:

> For your argument to have any cogency, I think it
> would need to be backed up by some theorems of the
> general form, "Let X be a consistent system with
> proof-theoretic ordinal alpha.  If alpha is frabjous,
> then X proves false theorems."

There is certainly no theorem of this form.  Just start
with your favorite base theory in second order arithmetic
and add the axiom "r is a well-ordering" for some recursive
well-ordering r of omega with order type alpha.  If the
axioms of the base theory are true and r actually is a
well-ordering, then all the axioms are true and hence the
formal system proves no false theorems.

But surely the more complex a formal system is, the less
likely it is to have some nice property like having a
model with a standard omega.  For a system like ZFC that
has a high degree of recursive complexity, in the sense
of having a large proof-theoretic ordinal, you would
have to assume it doesn't have a nice property like that,
unless you had some special reason to believe it did.


Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA

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