[FOM] : Questions on Cantor
Frode Bjørdal
frode.bjordal at ifikk.uio.no
Thu Jan 31 06:26:30 EST 2013
Thanks,
"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."
My compatriot Thoralf Skolem did this very precisely and satifactorily
around 1920. According to information related somewhere by Solomon
Feferman, according to my memory, Herman Weyl did so independently (and
unbeknownst to Skolem) around 1916.
Frode Bjørdal
2013/1/27 Irving Anellis <irving.anellis at gmail.com>
> 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),
> 207-246).
>
> 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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>
--
Frode Bjørdal
Professor i filosofi
IFIKK, Universitetet i Oslowww.hf.uio.no/ifikk/personer/vit/fbjordal/index.html
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