On 12/4/2013 12:12 PM, WM wrote: > Am Mittwoch, 4. Dezember 2013 15:47:46 UTC+1 schrieb fom:
>> >> >> Your statement speaks to what may be put under >> >> the scope of quantification. Cantor seems to have >> >> recognized fairly early that no transfinite notion >> >> could correspond with absolute infinity. It may >> >> have been his involvement with religious scholarship >> >> that helped him to avoid Russell's paradox directly. > > He did not avoid it. He knew fairly early, in the 1890s, that the set of all sets is impossible.
So, you would say that acknowledging the mathematical difficulty supported him in the theological arguments as opposed to the other direction.
Is that a correct assessment of your sense of things?
> Unfortunately he did not recognize that the set of all natural numbers that are followed by infinitely many natural numbers is impossible too. (I.e., the set of all numbers that is not all numbers.)
Well, it is perhaps simpler to view Cantor in this way given the state of modern mathematics. However, there had been no clear notion of topology in the sense that we have it today. The links to Aquinas which you posted clearly have him speaking of the kind of "possibility" of form that an object may take. It is not hard to see this in relation to the convergence of an open cover to a compactum.
Cantor's views can be understood as delineating some of these topological notions during a historical period committed to arithmetical and logical solutions at the expense of geometry.
> > The usual reply of matheologians, on the question how they know that they have all numbers without having the last one, is: There is no last natural number. But I im interested to see their escape with respect to the fact that no digit d_i ad no FIS is sufficient to accomplish a distinction of d from all rationals of the rationals-complete list. Obviously d contains somewhat more than every digit d_i. >
I must leave an answer to that question to Virgil. I am much too easily drawn into mistakes by someone with your skills. :-)