[FOM] Cantor and "absolute infinity"

[hendrik@topoi.pooq.com]

On Tue, Feb 21, 2006 at 06:48:19PM -0500, Harvey Friedman wrote:
> On 2/20/06 6:44 PM, "Arnon Avron" <aa at tau.ac.il> wrote:
> 
> > I agree with Friedman that all the principles of ZFC (except
> > infinity, of course), are obtained by extending principles
> > we know from the finite case. However, I believe that
> > understanding this would have made these people *more*
> > suspicious about ZFC, because the infinite case is so extremely
> > different from the finite one!
> 
> I claim that it makes the rest of the mathematical community very
> comfortable with ZFC, since they have learned intuitively that infinite sets
> do not cause any discernible problem with the overall picture.
> 
> Sure, they know that the picture cannot be the same at some level of detail,
> because infinite sets are different than finite sets.
> 
> However, the difference between infinite sets and finite sets only surfaces
> when one worries about functions. Somehow, people grasp that an infinite set
> is really like an extremely large finite set. So it is not plausible to them
> that there can be anything wrong with the same picture.
> 
> I certainly have spent decades thinking about how to say this. I have done
> work on transfer principles some years ago, and they are on my website, but
> they are not quite what we want.
> 
> I regard this as a very interesting and promising line in f.o.m. I am in the
> middle of an approach to this problem, and if all goes well, the results
> will be in a numbered posting.
> 
> > Now careless analogies are perhaps the main source of mistakes
> > and confusions in math.
> 
> We should think productively, and look for relevant transfer principles. In
> other words, make the analogy - which has apparently well satisfied
> virtually the entire mathematical community for perhaps 80 years - stronger
> by real theorems. 
> 
> >I am teaching set theory for many
> > years now, and each year I take (most often in vain)
> > great pain in warning my students against
> > making analogies between the finite case and the infinite one!
> 
> And I take great pains to tell students how strong the analogies are.
>  
> > Here are some examples that you know of course quite well:
> > 
> > 1) The self-evident principle that the whole is greater than its
> >  part (considered by Euclid to be a general axiom, not just
> >  a postulate) fails for infinite sets.
> 
> Only if you are thinking of cardinality.

Maybe cardinality is not such an intuitively sensible concept.

We seem to be interested in the existence or nonexistence of one-to-one 
maps between sets.  But when such a mapping does not exist, it may 
simply be because such a mapping happens to be absent from the universe.

If we consider a countable model, this is perhaps a bit clearer -- sets 
that are uncountable within the model can turn out to be quite countable 
seen fron the outside.

So I've started thinking of the issue as being one of intricate stucture 
rather than one of brute size, much as continuous maps between 
topological spaces may fail to exist even when they have equipollent 
sets of points.

Has anyone investigated what kinds of structure might be the relevant 
ones?  Structure of the sets themselves, or is it a kind of holistic 
property of the set theory (or its model(s)) that we have to consider?

-- hendrik