[FOM] : Questions on Cantor

Irving Anellis irving.anellis at gmail.com
Sun Jan 27 14:50:39 EST 2013

Set Theory and Its Role in Modern Mathematics_ (Basel/Boston/Berlin:
Birkhauser Verlag, 1999; 2nd revised ed., 2007) provides much more detail
at a high technical level the history of set theory from 1850 to 1940,
elucidating the connections of set theory and mathematical logic and
exploring the close interconnections between the history of set theory and
of real analysis.

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),

The concept of a proper subset in its modern usage appeared in Cantor’s
"Ein Beitrag zur Mannigfaltigkeitslehre (Journal fur die reine und
angewandte Mathematik 84 (1878), 242-258). However, the proposal to
distinguish between sets and classes was made by Zermelo in the final draft
for his Warsaw lecture of 1929 (see the "Vortrags-Themata für Warschau
1929/Lecture Topics for Warsaw 1929", in  Hans-Dieter Ebbinghaus, Craig G.
Fraser, & Akihiro Kanamori, editors), Ernst Zermelo, Collected
Works/Gesammelte Werke, Vol. I/Bd. I: Set Theory, Miscellanea/Mengenlehre,
Varia (Heidelberg/Dordrecht/London/ New York: Springer-Verlag, 2010,
374-389), esp. p. 325 in the introduction to this paper by Hans-Dieter
Ebbinghaus; see also Hans-Dieter Ebbinghaus, "Ernst Zermelo: A Glance at
His Life and Work", pp. 3-42, esp. p. 29) in the Collected Works; he may
well have had John von Neumann's 1925 "Eine Axiomatisierung der
Mengenlehre" in mind , which formalized Dmitry Mirimanoff's (1861-1945)
distinction between "ordinary" and “extraordinary" sets  ("Les antinomies
de Russell et de Burali-Forti et le probleme fondamental de la theorie des
ensembles”, L'Eseignement Mathematique 19 (1917), 37–52; "Remarques sur la
theorie des ensembles et les antinomies cantoriennes", L'Enseignement
Mathematique 19 (1917), 209-217, 21 (1920), 29-52, especially p. 42 of the
former) as well-founded and non-well-founded sets, that is (in GNB) between
sets and [proper] classes. The distinction between subsets and proper
subsets in ZF was enshrined by Abraham Adolf Fraenkel (1891-1965) in his
exchanges with von Neumann (see pp.  1-3 in Stanislaw Ulam, "John von
Neumann, 1903-1957", Bulletin of the American Mathematical Society 64
(1958, 1–49; reprinted in: John von Neumann (F. Brody & T. Vamos, editors),
The Neumann Compendium, vol. 1. (Singapore: World Scientific Publishing Co,
1995), xi-lix. 1–3), and esp. n. 3 on p. 10, quoting Fraenkel’s letter to
Stanislaw Ulam (1909-1984) on von Neumann's work in set theory). Von
Neumann in turn (in his "Eine Axiomatisierung der Mengenlehre", Journal fur
die reine und angewandte Mathematik 154 (1925), 219-240; English
translation by Stefan Bauer-Mengelberg & Dagfinn Follesdal in Jean van
Heijenoort (ed), From Frege to Godel: A Source Book in Mathematical Logic,
1879-1931 (Cambridge, Mass.,  1967), 393-413) 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. With this in mind, you might also want to
have a look at Gregory H. Moore's very rich and still highly informative
book, Zermelo's Axiom of Choice: Its Origins, Development, and Influence
(New York/Heidelberg/Berlin: Springer-Verlag, 1982).

Irving H. Anellis
Indiana University-Purue University at Indianapolis
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