[FOM] Cantor and "absolute infinity"

Harvey Friedman friedman at math.ohio-state.edu
Tue Feb 21 18:48:19 EST 2006

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

Cardinalities again.
> 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).

Again cardinalities.
> 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! Of course, a big mistake that one
could make is to reject the crucial enterprise because the very first
breakthroughs leave much to be desired. This is a very standard and
wrongheaded move we have all seen people make. This can discourage people
from continuing to improve the results.

> 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.

In any case, every justification in f.o.m. is "extremely weak". Finitists
think that any justification for even the bare beginnings of predicativity
is "extremely weak".

Ultrafinitists think that even the bare beginnings of finitism is "extremely

>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).

Stay tuned. 
> 1) If A is an hereditarily finite set than both A and P(A)
>  can effectively be listed (or well-ordered) ,

Not according to an ultrafinitist. Consider 10,000 versus 2^10,000. The
first is rather normal. Just look at the people in a small section of NYC.
The second "is absurd nonsense that is purely fictitious and cannot be
counted or realized or listed in any way, etc.". This is just the finitist
or predicativist complaint further down.

>and one can 
>  write a program which outputs all the elements of P(A) one by one.

Ultrafinitist AND applied computer scientists: the program can't run
properly. Don't you dare try to run it on my machine!

>  On the other hand, while N can effectively be listed

Ultrafinitist: this is absolute nonsense. You can't list it for lots of
reasons, including it's too big, and also you don't know what ... means.

>  and one can write a similar program for it, P(A) cannot
>  be effectively listed,

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.

>and no effective well-order for it
>  exists 

Set theorist: What does an effective well-order of P(N) mean anayway?
Effectivity usually refers to manipulating finite strings.

>(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?

>  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).

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?
> 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?

>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?

Harvey Friedman

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