[FOM] Re to Friedman on finite and infinite sets

[Arnon Avron]

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, 

You are the first mathematician I ever met who claims that he feels
comfortable with ZFC because of the analogies with finite sets.
Try to ask your medalist about it.

> However, the difference between infinite sets and finite sets only surfaces
> when one worries about functions. 

So what? Why should infinite sets be different in this 
respect from finite sets?

> Somehow, people grasp that an infinite set
> is really like an extremely large finite set. 

Even Galileo has not graspied it, and was surprised by the paradoxes of 
infinity. (Also no mathematician has doubted AC for finite sets,
and still there was a huge debate about it. Now everybody agrees
that for infinite sets one needs to add it as a NEW principle).
I really don't know on what ground you base this 
declaration of yours.

> make the analogy - which has apparently well satisfied
> virtually the entire mathematical community for perhaps 80 years - stronger
> by real theorems. 

I really don't know on what ground you base this
declaration of yours.
 
> >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.

I have strong doubt how well your students would have done
standard examinations on basic set theory in Tel-Aviv University.
  
> > 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.

So what? Why should infinite sets be different in this 
respect from finite sets?

> > 2) The trivial obvious fact that a+1 > a fails for infinite
> >  cardinalities.
> 
> Cardinalities again.

So what? Why should infinite sets be different in this 
respect from finite sets?

> > 3) the fact that every 1-1 function from a set A to an equipolent
> >  set B is unto B fails for infinite sets.
> 
> Cardinalities.

No. functions and relations. 
Why should infinite sets be different in this
respect from finite sets?

> > 4) The trivial identity a-a=0 is not only false in the infinite
> >  case. It is meaningless (still, students repeatedly "use" it).
> 
> Again cardinalities.

So what? Why should infinite sets be different in this
respect from finite sets?

> > 
> > What is clear is that
> > without  an objective, well-motivated  *very* good criteria
> > which analogies here
> 
> Let's roll up our sleeves and get to work! 

Fine. But I understand from this call that you admit that
the work has not been done yet. So again 
I really don't know on what ground you base your
declaration about the reason that mathematicians accept ZFC 
as the golden standard (assuming that this declaration is correct,
which I doubt. I believe most "core mathematicians" will not even
be able to tell what are the axioms of ZFC that 
they are supposed to accept as the golden standard).

> > are acceptable and which are not, your justification for Cantor
> > set theory or ZF is extremely weak.
> 
> So weak that almost the entire mathematical community has accepted it as the
> GOLD STANDARD for over 80 years.

Maybe they do, but not because of the justification you suggested,
which is very weak. Worse: at least until you do the work you
are calling for, your justification necessarily lead to horrible 
mistakes of students (this is not a speculation. It happens all the time).

> Set theorist: Oh, you are trying to simply tell us that P(N) is not
> countable? This result is not new, and is due to Cantor.

Of course. I am not trying to be original. I Just point out the 
implication of this fact to the analogy finite/infinite.

> >and no effective well-order for it
> >  exists 
> 
> Set theorist: What does an effective well-order of P(N) mean anyway?

In this case it suffices to understand it in an extremely weak sense
of "definable in some reasonable sense of definability".

> >(I don't understand in what sense a non-effective
> >  well-order exists, but this may be left to another discussion.
> 
> Set theorist: it exists by the axiom of choice. Since the axiom of choice is
> provable for finite sets, it is clear for arbitrary sets. Why should
> infinite sets be different in this respect from finite sets?

 Why should infinite sets be similar in this respect to finite sets?
(Note again that for finite sets we don't need AC as a new principle!)
 
> Set theorist: Since the power set axiom is provable for finite sets, it is
> clear for arbitrary sets. Why should infinite sets be different in this
> respect from finite sets?

I have given already very good reasons for it. You are not convinced.
I am not surprised. So do not be surprised that I am not convinced
by your arguments (and neither are, I believe, most mathematicians.
Try to ask them).

> > 2) If A is finite than the formula x=P(A) is absolute. In contrast,
> >  x=P(N) is not absolute.
> 
 What does absolute mean? Is this some technical notion in set theory?

Come on. For the present purposes I am using it in the technical
sense used in set theory (of course I personally might interpret it somewhat
differently, but this is irrelevant).

> 
> >For me this is a decisive reason
> >  to see the notion of the powerset of a *finite* set as transparent,
> >  while that of P(N) as based on assuming an infinite mind
> >  (like God or something like this).
> 
> So is the notion of the powerset of a set of cardinality n based on having
> an n mind ("like God or something like this"), where n = 2^2^2^2^1000?

I totally disagree, but I am not ready to make debates about it.
If you really think so (I doubt)  I can do nothing about it. 
I can only wonder why you are trying to give justification
to ZFC of any kind, if you dismiss any criticism by 
pointing out that anything can be criticized.

However, I'll return to this type of  post-modernist arguments
of yours in another posting.

Arnon Avron