[FOM] The lure of the infinite

[William Tait]

On Feb 16, 2006, at 5:27 AM, praatika at mappi.helsinki.fi wrote:

> Already Cantor himself made a distinction between infinite sets (of  
> various
> powers) which are dealt with in set theory, and absolute infinite,  
> which is
> too huge to be a set.

I think that "too huge" may misrepresent (or at least suggest a  
misrepresentation of) what Cantor meant in endnote 2 of his  
*Grundlagen* (1883) about the absolute infinite, as represented by  
the totality of transfinite numbers or the totality of powers. For  
this description suggests that these totalities have a definite size.  
I think that this is what he is rejecting. In his review of Frege's  
*Foundations of Arithmetic* he calls these totalities "quantitatively  
indeterminate." Thus for him (and Zermelo followed him on this) there  
is no well-defined universe of all sets, only an indeterminate one  
out of which we can carve greater and greater determinate parts---in  
this sense, we might say that it is a potential infinity. So for him  
there wasn't the embarrassment of absolutely proper classes, which  
are well-defined totalities but which somehow fail to be sets.

It is striking that Cantor;s theory of transfinite numbers therefore  
already, long before Goedel, contained the seed of incompleteness in  
set theory.

> 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.)

> This is also a comment on some earlier posting here:
> Although the first explicit occurrence of the interative concept of  
> set is
> in Zermelo 1930, is is arguable that Zermelo had the view implicitly
> already in 1908, and indeed that it can be traced back to Cantor  
> himself
> (or, so Hao Wang argued) - if so, this conception of sets was,  
> again, not
> threatened by the paradoxes.

There is some difficulty with understanding Zermelo on this, although  
I think that you are right. Cantor certainly had the notion of the  
iterated hierarchy of sets. In his 1890/1 paper in which he presented  
the diagonal argument, he notes that this leads to a hierarchy of  
powers cofinal with the the hierarchy of number classes. He clearly  
has in mind the hierarchy of sets obtained by iterating the powerset  
operation starting with the set of natural numbers.

Regards,

Bill Tait