[FOM] Cantor and "absolute infinity"

[Arnon Avron]

On Sat, Feb 18, 2006 at 11:35:47PM -0500, Harvey Friedman wrote:

> This issue is easily resolved if you look at set theory 
> as an extension of  finite set theory. 

...
 
> I wouldn't call these people [Hilbert, Dedkind, Weyl] "stupid", 
> but if they understood the proper
> analogy with finite set theory, they would see that the 
> paradoxes do not  threaten Cantor's theory at all.

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!

Now careless analogies are perhaps the main source of mistakes 
and confusions in math. 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!

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.

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

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

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

I can go on and on with this. What is clear is that
without  an objective, well-motivated  *very* good criteria 
which analogies here 
are acceptable and which are not, your justification for Cantor
set theory or ZF is extremely weak.  and I am not aware of
any such criterion (by the way, in your outline of the way
you explain set theory to novice I noted that you have stopped
short before reaching the "paradoxes of infinity". I am not
wondering why).

  Two specific more sophisticated differences between finite set theory
and general set theory which for me are crucial:

1) If A is an hereditarily finite set than both A and P(A) 
   can effectively be listed (or well-ordered) , and one can 
   write a program which outputs all the elements of P(A) one by one. 
   On the other hand, while N can effectively be listed
   and one can write a similar program for it, P(A) cannot 
   be effectively listed, and no effective well-order for it
   exists (I don't understand in what sense a non-effective
   well-order exists, but this may be left to another discussion.
   Here it suffices to note that Zermello's proof of his well-ordering
   theorem relied on the powerset axiom - an axion I find as very 
   suspicious).

2) If A is finite than the formula x=P(A) is absolute. In contrast,
   x=P(N) is not absolute. 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).

Arnon Avron

 
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