[FOM] Cantor and "absolute infinity"

Arnon Avron aa at tau.ac.il
Sat Feb 18 18:09:25 EST 2006

On Thu, Feb 16, 2006 at 10:18:34PM -0600, William Tait wrote:
> > So there is a sense in which paradoxes never really
> > threatened Cantorian set theory, but only Frege's system, which is  
> > not the
> > same thing.
> Exactly so. But unfortunately, Cantor buried his remarks about the  
> absolute infinite in an endnote. Frege refers to Cantor's paper in  
> such a way that it is clear that he read the description of the  
> transfinite numbers. One may wonder whether, if Cantor had positioned  
> his remarks on the absolute infinite more prominently, Frege might  
> have been saved dome pain and people such as Dedekind, Hilbert and  
> Weyl would have seen the paradoxes of set theory simply as mistakes.  
> Maybe the history of foundations in the early 20th century would have  
> been quite different. (Purkert has suggested that Cantor buried these  
> remarks in an endnote because he wanted to sell his theory and did  
> not want people to confront initially the complication of the  
> absolute infinite. I want that to be false.)

I don't buy this story and speculation. I cannot say for sure what
was Cantor's motivation, but I can say for sure that he had
good objective reasons to bury his vague remarks. Otherwise the
obvious question would have been (and was later):
what is the criterion that distinguishes between "absolutely
infinite" collections and non-absolute infinite collections 
(= infinite "sets")? After all, the comprehension schema
*was* used by Cantor whenever it suits him, and (as far as I know)
he has never given definite conditions when it can be applied 
and when it cannot. Thus he took the powerset of a given set A
to be a "set". Did he have any characterization of this "set",
except as the  set of all sets which are subsets of A 
(a clear application of unrestricted comprehension)? 
Had Cantor positioned his remarks on the absolute infinite more 
prominently, he would have had no choice but to give 
a convincing argument why P(N) is a set
and not an absolutely infinite collection (as the fact that 
it is not countable could have suggested!). I cant see what
this argument could have been. In fact, I cant see what 
criterion Cantor could have had for telling what collections
are "absolutely infinite" (and according to which
P(N) is not "absolutely infinite") except to ban collections/sets
the assumption of which leads to contradictions...

  Again: I have not made a historical research, but I am quite
sure that Cantor's talks about absolute infinity were caused
by Cantor's paradox, a paradox that occurs to every good student who 
learn for the first time Cantor's theorem (that the powerset of 
A has bigger cardinality than that of A). An immediate question is: 
what if A is the set of all sets? Cantor certainly should have asked
himself the same question, and I guess the only way out he
has found was to say that this
collection is not a set. But why? here come the 
vague idea of "absolute infinity".  I am pretty sure that
this idea was *forced* on Cantor by the contradictions, and not
that he had never seen any. Moreover, I believe that had cantor 
in his possessions a convincing theory here, he would have given it,
rather than "bury" it. 

The upshot is: Hilbert, Dedkind, Weyl etc were not so stupid
after all, when they saw that the paradoxes are threatening
Cantor's theory too, and not only Frege's. What is problematic
for both is the comprehension principle. No theory in which 
infinite sets may be objects can exist without some form of it, 
because we simply have no other method to describe (or refer to
or introduce) specific infinite sets. Unfortunately, this
principle leads to contradictions. Being aware of Cantor's remarks
would not have changed this basic situation. 

I think that Hilbert, Dedkind, Weyl etc deserve more credit than
thinking that if only they knew Cantor's hidden views they would never
have seen problems in set theory. They were independent
thinkers, and the mathematical theory was available for them
to judge objectively  whether its foundations are secure. I cant 
imagine any of them thinking "Aha, Cantor
meant only sets that are not absolutely infinite! Why has nobody 
told me before, and how come I could not have understood it myself?").

Arnon Avron

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