[FOM] Arithmetical soundness of ZFC (non-platonic)
nweaver at math.wustl.edu
Wed May 27 15:49:04 EDT 2009
Most of the discussion about arithmetical soundness of ZFC has
not relied on any platonist assumptions. My overall impression
is that the proposed reasons for believing ZFC is sound have
been very weak and indirect.
Harvey Friedman suggests that we "Continue proving from big set
theories, various arithmetical theorems. Then wait and see if
they are later proved using `innocent' systems such as PA or
even EFA". I just don't see what this has to do with soundness.
Some arithmetical results provable in ZFC will also be provable
in weaker systems and some will not. If ZFC is consistent these
are the only two possibilities. Does the existence of the
latter count as evidence against soundness? I just don't see
the connection. What I do see is that "natural" number-theoretic
results tend to be provable in weaker systems, whereas the
results that need heavy set theory tend to appear contrived, to
me anyway. If anything, that looks like evidence against the
arithmetical soundness of ZFC (but not very strong evidence).
Joe Shipman makes the interesting suggestion that all consistent
extensions of ZFC that have been seriously proposed may be
arithmetically compatible. Simultaneously John Steel comments
that NF could be consistent yet arithmetically incompatible with
ZFC. My first reaction is that I'm not sure how much effort has
been put into determining whether various possible set theories
are arithmetically compatible. There's also a problem that the
theories that have been studied the most tend to fall within the
mainstream set theory tradition, have related motivations, and
thus don't really seem like independent examples. But still, I
will say that at some point this does become the kind of evidence
that I would take seriously. It's still pretty indirect, though.
I think the most we could say, if Joe's suggestion completely
panned out, is that it would tilt the balance in favor of "ZFC is
probably arithmetically sound".
Monroe Eskew notices that all we need for ZFC to be sound is a
model with a standard omega, and thinks it is likely that at
least some models of ZFC do have a standard omega. He may not
have read the point I made earlier (in message # 013645) that
we expect a generic theory to be consistent but not arithmetically
sound. Soundness is a very rare, special property, a much much
stronger property than consistency, so if all we knew about ZFC
were that it is consistent, we would have to assume it probably
does not have a model with a standard omega. You don't need a
special reason to lack a model with a standard omega, you need
a special reason to have such a model.
Conspicuously absent from the non-platonic side of the discussion
has been any suggestion about how we could construct a model of
ZFC with a standard omega. Now, I am not a finitist. I accept
transfinite constructions. So if ZFC is consistent I know how
to build a model. But I don't have any idea how I could come up
with a model with a standard omega.
Is there any known method for constructing such a model that
doesn't assume a preexisting platonic world of sets? If not,
then the non-platonic defense of ZFC being arithmetically sound
amounts to an assertion that a particular mathematical object
(a model of ZFC with a standard omega) exists, without saying
how it could be constructed, in the mainstream mathematical
sense that allows transfinite constructions. The only reasons
to think it exists are these very indirect arguments that are
basically aesthetic in nature.
St. Louis, MO 63130 USA
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