[FOM] : Questions on Cantor

Christopher Menzel cmenzel at tamu.edu
Wed Feb 13 13:39:49 EST 2013

On Jan 27, 2013, at 1:50 PM, Irving Anellis <irving.anellis at gmail.com> wrote:
> ...With respect to the specific question of well-founded sets in Cantor, one should look to his distinction between complete and incomplete multiplicities, especially to his  "Beitrage zur Begrundung der transfiniten Mengenlehre, Theil II", Mathematische Annalen 49 (1897), 207-246).

I take it that the distinction between complete/incomplete multiplicities here is the one Cantor also referred to as the distinction between consistent/inconsistent, alternatively, determinately infinite/absolutely infinite multiplicities. But I don't find much discussion of that distinction in the Beiträge (Jourdain alludes to it in the introduction and cites a well-known quote from Cantor (from an earlier work) about the absolutely infinite in a footnote) and, in any case, I'm not sure how the distinction bears on well-foundedness, as the distinction arises independently of foundation. Can you expand on what you have in mind?

> …The distinction between subsets and proper subsets in ZF...

I take it this should read "sets and proper classes".

> ...was enshrined by Abraham Adolf Fraenkel (1891-1965) in his exchanges with von Neumann....Von Neumann in turn clarified and formalized Zermelo's distinction between sets which are "definit" and those which are not or have Definitheit and those which do not; that is, a proper class is a set having a definite property.

I had thought that the issue in the debate over definite properties concerned Zermelo's original formulation of the axiom of separation. So again I don't see the connection to the set/proper class distinction.

Chris Menzel

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