Abstract
Russell tells in his 1903 “The principles of mathematics” (POM p 101) that he “was led to it [his paradox] in the endeavor to reconcile Cantor’s proof that there can be no greatest cardinal number with the very plausible supposition that the class of all terms … has necessarily the greatest possible number of members”. Cantor’s proof mentioned here is the proof of Cantor’s Theorem (1892) which, Russell says (p 362), “is found to state that, if u be a class, the number of classes contained in u is greater than the number of terms of u”. The class of all terms thus appeared to Russell as a refutation to Cantor’s Theorem. Russell’s Paradox was thus obtained within a Lakatosian proof-analysis.
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Notes
- 1.
On the chronology of Russell’s Paradox see Anellis 1991. Regarding the good reasons to call this paradox the Zermelo-Russell paradox, see Ebbinghaus 2007 p 45 ff. Regarding Zermelo’s preference of the term ‘antinomy’ to ‘paradox’ see ibid. footnote 62.
- 2.
‘Class’ is Russell’s term for Cantor’s ‘set’ and is explicated in Chap. 6 of POM. ‘Term’ is a technical term with Russell, rather loosely defined to stand for anything that may occur in thought, language or reality (“can be counted as one”) (POM p 43, 55); it corresponds to Dedekind’s ‘thing’ (1963 p 44). The class of all terms we denote by U. Occasionally Russell juggles with other collections in place of U such as the class of all objects (Grattan-Guinness 2000 p 320) or the class of all propositions (POM p 367; cf. Russell 1919 p 136, Russell 1944 p 388). The class of all classes, which is clearly the class of all subclasses of U, we denote, following Whitehead 1902, by Cls.
- 3.
Russell describes here a clash of two gestalts: under one gestalt there is no maximal set; under the other there is such a set.
- 4.
The name ‘Cantor’s paradox’ is used for this contradiction (as well as for the contradiction regarding the class of all cardinal numbers). Moore-Garciadiego (1981 p 331; cf. Peckhaus 2002 p 5) reference POM p 367 as the place of origin of the name, but there, only the expression ‘Cantor’s argument’ appears. The issue is clarified in Grattan-Guinness 2000 p 310f, 312f.
- 5.
The relata of x (in u) is the class of all members y of u such that xRy – p 24.
- 6.
It seems that if ω is empty the contradiction should emerge from y ∈ ω.
- 7.
We remind the reader that Dedekind originally defined his chains for many-one mappings (see Sect. 9.1).
- 8.
Two classes are similar (POM p 113) when there is a “one-one relation whose domain is the one class and whose converse-domain [or ‘range’ in prevailing terminology] is the other class”. ‘Similar’ corresponds to Cantor’s ‘equivalent’ but it implies a 1–1 relation whereas equivalence implies a 1–1 mapping.
- 9.
We have here an example of an application of CBT to inconsistent sets.
- 10.
Dedekind’s proof (see Sect. 9.2) contained such a construction, but it was published only in 1932. Zermelo’s proof, predating Peano’s paper, also contained a similar construction but in Poincaré’s rendering of it (see Sect. 19.5), published in the same month when Peano’s paper was published, the construction was not stated explicitly.
- 11.
Here Russell applies, tacitly or without notice, the generalized Cantor’s theorem.
- 12.
Actually the class omitted is the class of all terms that do not belong to their image under the patched mapping. Since all the terms that are not classes belong under this mapping to their image, the omitted class is indeed composed of all classes that do not belong to their image, namely, to themselves.
- 13.
Move from the conditions of our version to those of Bunn’s version requires AC.
- 14.
The latter is referenced as 1978.
- 15.
Plural is used here because Russell did not rule out the possibility that the hidden lemma could be found in one of the lemmas that served to establish Cantor’s theorem.
- 16.
Russell refers here to the formulation of Cantor’s theorem stated at the beginning of this chapter.
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Hinkis, A. (2013). The Role of CBT in Russell’s Paradox. 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_16
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