FOM: Charles Silver on silly questions
Charles Silver
csilver at sophia.smith.edu
Mon Oct 12 11:25:42 EDT 1998
On Mon, 12 Oct 1998, John Mayberry wrote:
> There is nothing wrong with the proof that all models of these
> second order axioms are isomorphic (we don't have to talk about the
> "standard structure" here). But for the proof to get off the ground we
> must assume that there is a transfinite set with a power set.
I don't see that you need a *transfinite* set.
> Of course
> most mathematicians simply take that assumption for granted. But if you
> don't take it for granted - and clearly Sazonov doesn't - then you are
> left with the problem of how to lay the foundations for natural number
> arithmetic.
I don't see that this involves "formal foundations." I think the
ideas work on an informal level. Sure you need something like sets. But,
I'm not saying you need *sets*, just something a little like them. And,
as I said above, I don't see that anything about a *transfinite* set is
needed at all.
> And in those circumstances Sazonov is right: the problem is
> not that it is *untrue* that each natural number is obtained by
> starting from 0 and iterating the successor function a finite number of
> times;
> it is rather that it is not clear what such a claim means.
I'm not with you here. Which claim is not unclear?
> Dropping Dedekind's infinitary assumptions deprives us of his
> mathematically precise analysis of the notion of finite iteration (his
> theorem on definition by induction in Section 126 of his essay, and the
> discussion in Section 130).
I don't see that this stuff is needed. I see that we *do* often
make various assumptions about infinity, and so forth, but I don't see
that we *have* to. I think the proof still holds without having
transfinite sets.
On a (I think) different and more complicated level, I suspect
that the ideas behind Steve Simpson's work in reverse mathematics could
shed some light on this, but I'm not sure whether the technical
elucidation of those ideas resolve the matter. The problem I see is that
if you prove 'P -> Q', where P incorporates a number of assumptions and Q
is some mathematical result, the temptation is to think that P is
*required* for Q. That is, the result 'P -> Q' can make you think that P
is *necessary* for Q, when it can easily be the case that, say, 'R -> Q'
as well. Steve's response to this implicit criticism would, I think, be
that P represents a system with a certain degree of power and he can
buttress his claim that not only is P sufficient but also necessary by
showing that anything weaker than P does *not* yield Q. Hence R,
according to Steve (according to my interpretation of what he'd say) has
to be at least as strong as P. The fault with this that I think Vladimir
Sazonov would find is that both P and R are still formalizable within a
metatheory that he finds questionable. But, I am not sure exactly what he
finds questionable. I think his main complaint has something to do with
formalization. And, I don't think formalization is the real issue.
Charlie Silver
More information about the FOM
mailing list