Abstract
In the late 1890s a new gestalt for the presentation of the comparability of sets emerged. Following Fraenkel (1966 p 72f) we call it the scheme of complete disjunction. It was published first by Borel in his 1898 book, in the appendix where he brought Bernstein’s proof of CBT. It then appeared in two letters of Cantor from 1899, to Schoenflies and to Dedekind, and in Schoenflies’ report of 1900. The scheme is noteworthy because it brought logical analysis, of the propositional calculus kind we used in Sect. 5.4, to what seems to be a pure set theoretic context. We will argue that it was Cantor who developed the scheme, following an analysis of Schröder (1896).
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Notes
- 1.
Fraenkel may have picked the name from Schröder’s (1898 p 346) “complete development” (vollständigen “Entwickelung”).
- 2.
Seemingly, Schoenflies was not aware of the intricacies regarding inconsistent sets and cardinal numbers discussed in the previous chapters, even though the subject was hinted in the letter he speaks about.
- 3.
Schoenflies clearly attributes the scheme to Cantor. The letter he mentions is the one of June 28, 1899. It seems that in Göttingen, by 1899, they have not yet received Borel’s book of 1898.
- 4.
The scheme is used in Kant’s division to analytic-synthetic, a priori-aposteriori. So what Shoenflies must have meant is the use of the scheme for mathematical statements. Still, we doubt it that complete disjunction was first applied for the analysis of concepts by Cantor or in Cantor’s time. In 1906 Hessenberg refers to the scheme (p 495) as “the familiar logical scheme” of “fourfold disjunction”, which, however, could mean the use of the scheme in philosophical discussions.
- 5.
In addition, we can imagine that the “scandalous” nature of incomparable sets implied by the fourth case, gave the scheme a special flavor which fitted well the air of decadence of the fin du siècle.
- 6.
- 7.
Here Cantor added a footnote that when he speaks of a subset he always means a proper subset because Dedekind’s “subset” included the case when the subset is equal to the set (see Zahlen p 46; see footnote to Sect. 3.2).
- 8.
The scheme to Dedekind differs from the scheme to Schoenflies in that there the subsets are named in all the cases and here only in cases II, III. Also there the order of addressing M and N is interchanged, except in case II. Finally, here the clause ‘(or even more such)’ is omitted.
- 9.
We find it intriguing to observe the somewhat obsessive tendency, shared probably by us all, to attempt to put in optimal arrangement a pack of discrete elements with symmetrical or antisymmetrical characteristics.
- 10.
Because of this statement, Mańka-Wojciechowska (1984 p 192) understood Cantor’s words to imply that he was not interested in a direct proof of CBT.
- 11.
Russell, who was a realist too, had no problem in the acceptance of a tentative approach to the foundations of mathematics, and he was eventually joined by Gödel (1944 p 449).
- 12.
Actually it is A*-E but Schröder was not aware of the difference between the comparability of cardinal numbers and the comparability of sets.
- 13.
Schröder uses = for our ↔, juxtaposition for &, and + for |.
- 14.
Note that the first disjunct applies to infinite sets while the second to finite sets.
- 15.
Let us call these assertion F> and F<, or plainly F. Thus not only A-E are necessary to establish P but F too; this is clear in the 1898 argumentation.
- 16.
Actually it is A* but Schröder did not differentiate the two.
- 17.
Schoenflies does mention the 1898 paper too as ‘cf.’ to the 1896 source.
- 18.
Jaresbericht der Deutschen Mathematiker-Vereinigung.
- 19.
- 20.
Incidentally, Schröder’s 1898 races with Burali-Forti’s papers of the period to be the first paper on Cantor’s set theory of the 1895 Beiträge and the first paper to apply symbolic logic to Cantorian set theory.
- 21.
Note that there is a mistake in the text of Peckhaus’ paper to which the referenced footnote relates: the year of the conference appears as 1894 instead of 1896. The mistake is repeated in Ebbinghaus 2007 p 90.
References
Borel E. Leçons sur la théorie des functions, 1950 edition, Gauthiers-Villars: Paris; 1898
Borel E. Sur les ensembles effectivement énumèrables et sur les definition effectives. Rendiconti della Reale Accademia dei Lincei (Roma) (5)28 II(1919), 163–5.
Bunn R. Developments in the foundation of mathematics from 1870 to 1910. In: Grattan-Guinness I, editor. From the calculus to set theory, 1630–1910: an introductory history. Princeton: Princeton University Press; 1980.
Cantor G. Gesammelte Abhandlungen Mathematischen und philosophischen Inhalts, edited by Zermelo E. Springer, Berlin 1932. http://infini.philosophons.com/.
Cavailles J. Philosophie mathématique. Paris: Hermann; 1962.
Dauben JW. Georg Cantor. His Mathematics and the Philosophy of the Infinite, Cambridge MA: Harvard University Press; 1979. Reprinted by Princeton University Press, 1990.
Dauben JW. The development of Cantorian set theory. In: Grattan-Guinness I, editor. From the calculus to set theory, 1630–1910: an introductory history, Princeton university press; 1980.
Ferreirós J. Labyrinth of thought. A history of set theory and its role in modern mathematics. Basel/Boston/Berlin: Birkhäuser; 1999.
Fraenkel AA. Abstract set theory. 3rd ed. Amsterdam: North Holland; 1966.
Gödel K. Russell’s mathematical logic. In: Benacerraf P, Putnam H, editors. Philosophy of mathematics: selected readings. 2nd ed. Cambridge: Cambridge University Press; 1983. p. 447–69.
Grattan-Guinness I. The rediscovery of the Cantor-Dedekind correspondence. Jahresbericht der Deutschen Mathematiker-Vereiningung. 1974;76:104–39.
Grattan-Guinness I. The search for mathematical roots, 1870–1940: logics, set theories and the foundations of mathematics from Cantor through Russell and Gödel, Princeton University Press; 2000.
van Heijenoort J. From Frege to Gödel. Cambridge, MA: Harvard University Press; 1967.
Mańka R, Wojciechowska A. On two Cantorian theorems. Annals of the Polish Mathematical Society, Series II: Mathematical News. 1984;25:191–8.
Medvedev FA. 1966. Ранняя история теоремы эквивалентности (Early history of the equivalence theorem), Ист.-мат. исслед. (Research in the history of mathematics) 1966;17:229–46.
Meschkowski H, Nilsen W. Georg Cantor: briefe. Berlin: Springer; 1991.
Peckhaus V. Ernst Schröder und die ‘pasigraphischen Systeme’ von Peano und Peirce. Modern Logic 1990/1991;1:174–205.
Schoenflies A. Die Entwicklung der Lehre von den Punktmannigfeltigkeiten, I, Jahresbericht der Deutschen Mathematiker-Vereinigung 1900;8.
Schoenflies A. Zur Erinnerung an Georg Cantor. Jahersbericht der Deutschen Mathematiker-Vereinigung. 1922;31:97–106.
Schröder E. Über G. Cantorsche Sätze. Jaresbericht der Deutschen Mathematiker-Vereinigung. 1896;5:81–2.
Schröder E. Über Zwei Defitionen der Endlichkeit und G. Cantorsche Sätze, Nova Acta. Abhandlungen der Kaiserlichen Leopold-Carolinschen deutchen Akademie der Naturfoscher. 1898;71:301–62.
Zermelo E. Neuer Beweiss für die Möglichkeit einer wohlordnung, Mathematische Annalen 1908a;65:107–28. English translation: van Heijenoort 1967;183–98.
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Hinkis, A. (2013). The Scheme of Complete Disjunction. In: Proofs of the Cantor-Bernstein Theorem. Science Networks. Historical Studies, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0224-6_6
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