[FOM] Cantor and "absolute infinity"

[William Tait]

On Feb 18, 2006, at 5:09 PM, Arnon Avron wrote:

>>  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.
>

My only story is the assertion about the content of an endnote in  
Cantor's *Grundlagen*, published in 1883. So it is a 'story' only in  
the sense of a 'history'; it is accurate and I wasn't trying to sell  
it. Please read the endnote.

My speculation is that if people had taken in what Cantor said in  
endnote 2, namely that there are totalities, such as the totality of  
transfinite numbers and the totality of powers, which are not sets,  
the subsequent history (story) might have been substantially  
different. Again, I wasn't trying to sell it. Indeed, I'm not  
convinced of it, but it seems still to me very plausible.

> 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")?
>

Why do you call the discussion in endnote 2 vague? It states quite  
clearly that the totality of transfinite numbers or powers is not a  
set. He does not give an argument for this in the case of the  
transfinite numbers; but there is a simple one: Every set of  
transfinite numbers has least upper bound. So if the totality of   
numbers is a set, then it has a lub alpha. But then alpha < alpha.  
The argument in the case of powers is that they are in one-to-one  
correspondence with the transfinite numbers.

> 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)?
>

My goodness! Whenever it suits me I use the principle that every real  
number has an inverse: It suits me, however, only when the real is  
none-zero.

In the third paper in the series entitled "On infinite linear  
pointsets" (the Grundlagen is a slightly expanded version of the  
fifth paper in that series) he did state the comprehension principle  
in this form: In any conceptual sphere, any collection of  objects of  
that sphere that is defined by a well-defined property (in that  
sphere) is a set. (There is a further complication in the definition;  
but it is not relevant to this discussion.)  The conceptual spheres  
he was referring to in mathematics were explicitly stated (later on  
in the paper) to be arithmetic, function theory and geometry. In this  
context, there is nothing contradictory in the comprehension principle.

In the Grundlagen, he modified the principle, adding the---indeed  
vague---condition that the collection "can be thought of as one."  
But, however one understands this, it does not admit the general  
comprehension principle. For he explicitly rejected the application  
of that principle to the concept "x is a transfinite number".


> 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.
>

My remarks were about Cantor's Grundlagen, written in 1883. He had at  
that time not introduced the power set operation. He did that in his  
paper of 1890/1 in which he introduced the diagonal argument. Of  
course, he did regard the reals as forming a set. In fact, he had a  
conjecture about that in the paper: He thought they are in one-to-one  
correspondence with the second number class.

> 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...
>

Leaving aside the question of P(N), of course I agree. The  
contemporary search for new axioms for set theory seeks to replace  
this negative criterion by positive ones.

> 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).
>

By Cantor's paradox I assume that you mean here the application of  
his diagonal argument to obtain Russell's paradox. But then you offer  
an excellent example of the fallibility of states of being "quite  
sure":  Again, Cantor \wrote about the absolute infinite in 1883. He  
had not at that time yet discovered the diagonal argument.


> 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.
>

No, it was not a question that he would have asked in the framework  
of the Grundlagen. But such a universe would include all the number  
classes. His answer then that it is not a set would follow from the  
fact that it is not in one-to-one correspondence with a number class.  
That seems to me to be a really good way out.

But enough. As you say, you have not made the historical research. I  
yield to no-one in my taste for a priori history ("It should have  
gone down that way, therefore it did"); but, alas, Arnon, sometimes  
you just have to read the book.

Kind regards,

Bill Tait