Abstract
When seen in the “correct” light, the contradictions of set theory are by no means disastrous, but instructive and fruitful. For instance, the antinomies of Russell and Burali-Forti live on in the systems of axiomatised set theory in the guise of established theorems. Zermelo used the Russell-Zermelo argument to prove that every set possesses a subset which cannot be an element of that set, and from which it follows that there can be no universal set ((Zermelo, 1908b, pp. 264–265), p. 203 of the English translation), and the essentials of the Burali-Forti argument can be used to prove that there is no ordinary set of all (von Neumann) ordinals.
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Notes
- 1.
The Burali-Forti argument is explicitly used in this way in (von Neumann, 1928, p. 721), though the argument was clearly known to von Neumann much earlier, since all the essentials are present in von Neumann’s new account of the ordinals from the early 1920s (see von Neumann, 1923), and this form of the Burali-Forti argument is explicitly mentioned by von Neumann in a letter to Zermelo from August 1923. (See Meschkowski, 1966, pp. 271–273.) Zermelo had given the definition of the “von Neumann” ordinals by 1915, and possibly as early as 1913. In a lecture course in Göttingen in the Summer Semester of 1920 entitled “Probleme der mathematischen Logik,” Hilbert shows (pp. 15–16) how the Burali-Forti Paradox can be reproduced in the framework based on Zermelo’s definition: if W is the set of all ordinals, then W would itself be an ordinal according to the definition, and must therefore be a member of itself, contradicting one of the central theorems about these ordinals. The lectures can be found in Chapter 2 of Ewald and Sieg (2011). Despite Zermelo’s precedence here, von Neumann is still the real discoverer of the von Neumann ordinals, for he was the first to give a complete presentation of the relevant theoretical material, and in particular to recognise the importance of the Axiom of Replacement.
- 2.
- 3.
- 4.
In this paper, and elsewhere, Russell uses the term “proper class” where we would now use the term “set”.
- 5.
The analogy with Zermelo’s argument that the universe is not a set is clear. Russell is close to isolating the notion of non-absoluteness, though his condition is stronger; a property ψ is absolute when it is possible to find at least one V such that \(\{x : \psi(x)\}^{V} \neq \{x : \psi(x)\}\), not that this necessarily holds for all V.
- 6.
For the remark about “current logical assumptions,” see (Russell, 1907, p. 37). Gödel points out (Gödel, 1944, p. 135) that there are actually three distinct formulations of the VCP relied on in Russell’s writings. For Gödel’s discussion of these, see op. cit., pp. 455ff. Goldfarb suggests in (Goldfarb, 1989) that for Russell these formulations may be more intimately connected than Gödel’s discussion allows.
- 7.
†When I say that a collection has no total, I mean that statements about all its members are nonsense
- 8.
Russell (1908, p. 225). The relation between the “self-reproductive” properties isolated by Russell and the VCP was well summed up by Gödel:
I mean in particular the vicious circle principle, which forbids a certain kind of “circularity” which is made responsible for the paradoxes. The fallacy in these, so it is contended, consists in the circumstance that one defines (or tacitly assumes) totalities, whose existence would entail the existence of certain new elements of the same totality, namely elements definable only in terms of the whole totality. This led to the formulation of a principle which says that no totality can contain members definable only in terms of this totality [vicious circle principle]. ((Gödel, 1944, p. 133). The square brackets are in the original.)
- 9.
- 10.
- 11.
Poincaré (1912, p. 4, 1913a, pp. 87–88), pp. 67–68 of the English translation. The mention of drawers in this passage recalls what Poincaré has to say in his earlier paper from 3 years before; see p. 194 above. It is very likely that Poincaré distinguishes between the “extensional” and the “comprehension” views precisely because Zermelo hints at such a distinction in the section of (Zermelo, 1908a, pp. 117–118) which replies to Poincaré. See also (Hallett, 2010, pp. 109–112).
- 12.
Something like the IST is pursued in Martin-Löf’s constructive type theory; see e.g., (Nordström et al., 1990, p. 27).
- 13.
This resolution of the paradox was pointed out by Skolem in his original paper, (Skolem, 1923, p. 223).
- 14.
In what follows, we will slip rather sloppily between talk of formulae, the domains they determine, and the extensions of the formulae, thus sets or proper classes.
- 15.
See (Kunen, 1980, pp. 118–119).
- 16.
Whereas it is quite straightforward what \(\sigma^\varphi\) means in ZF, speaking strictly “\(< \varphi, \in\upharpoonright\negthickspace\varphi >\)” makes no sense, since the extension of ϕ might not be a set. However, this abuse of notation is of a piece with the sloppiness pointed out in n. 13.
- 17.
This notion of absoluteness is taken from (Bell and Machover, 1977, p. 502).
- 18.
See See (Kunen, 1980, p. 134ff.).
- 19.
- 20.
This seems to provide some link between the notions of absoluteness and impredicativity. See (George, 1987) for references to the idea that impredicativity concerns unbounded quantifiers. The link between the two notions is implicit in Poincaré’s analyses of the antinomies given in § 2. But as George points out, this cannot be all there is to the notion, which seems irredeemably imprecise.
- 21.
Let us note in passing that, in general, axiom systems are not the simple sum of their axioms, but that the axioms (so to speak) co-operate. Thus, although apparently weak on its own, the power-set axiom is enormously powerful when combined with the Axioms of Separation (or Replacement) and Infinity.
- 22.
This way of putting the matter is taken from (Bell and Machover, 1977, p. 509).
- 23.
See (Bell and Machover, 1977, pp. 480–481).
- 24.
For clear accounts of the proof of AC from \(\mathrm{ZF} + \, \mathrm{V} = \mathrm{L}\), as well as Gödel’s main “reducibility lemma,” see (Kunen, 1980, pp. 174–175) or (Bell and Machover, 1977, pp. 517–522). Bell and Machover point out the similarity of the proof of the main lemma with the proof of the non-absoluteness of P(x); see p. 522.
- 25.
It is worth pointing out that it is the iteration of D(x) through the classical ordinals that prevents the constructible hierarchy of the L α being an appropriate setting for predicative mathematics. See for example (Kreisel, 1960, p. 386). According to what Kreisel calls “the fundamental idea of predicativity” (ibid., p. 387), an ordinal α is predicatively legitimate for use in defining a given level Σ α of predicative definitions if there is a lower level Σ β (with \(\beta < \alpha\)) such that there is a well-ordering of the natural numbers of type α definable by a formula from \({\Sigma}_{\,\beta}\). (See also (Gödel, 1944; Wang, 1954).) This is not in general true of the L-hierarchy.
- 26.
In his first paper on the axiomatisation of set theory (Zermelo, 1908b, p. 262), Zermelo says the following:
Set theory is concerned with a “domain” \(\,\mathfrak{B}\,\) of objects, which we will call “things”, and among which are the sets.
- 27.
- 28.
Skolem (1923, p. 229) (English translation, pp. 298–299).
- 29.
- 30.
As Cohen himself said (Cohen, 2005, p. 2417):
For example, he [Skolem] pointed out the existence of countable models of set theory. … But certainly he was aware of the limitations on what can be proved. In a remarkable passage, he even discuses how new models of set theory might be constructed by adding sets having special properties, although he says he has no idea how this might be done. This was exactly the starting point for my own work on set theory, although I was totally unaware that Skolem had considered the same possibility.
For a more detailed discussion, see (Kanamori, 2008).
- 31.
A chain in a partially ordered set p is a subset of p in which the ordering relation is total. Two elements x,y of p are compatible if there is an element z of p such that \(z\leq x, z\leq y\), and incompatible if there is no such z. An antichain in p is a subset q of p such that any two distinct elements of q are incompatible. The c. c. c. for p then says that any antichain of p is countable. See (Kunen, 1980, p. 53)
- 32.
See (Kunen, 1980, pp. 204–208).
- 33.
The material in this paper has been through many incarnations over the last 20 years. The inspiration for it, though, is John Bell, who first made me aware of the internalised version of the Skolem Paradox. For discussions on this and related matters, I am also deeply indebted to the late George Boolos, William Demopoulos, Michael Friedman, Moshé Machover, the late John Macnamara, Mihaly Makkai, Stephen Menn and Gonzalo Reyes. As acknowledged in n. 3, I also owe a substantial intellectual debt to Wilfrid Hodges. I am also very grateful for the generous support of the Social Sciences and Humanities Research Council of Canada over many years, as well as the FQRSC of Québec, formerly FCAR. It is also a pleasure to acknowledge the gracious support of the Alexander von Humboldt Stiftung and the Akademie der Wissenschaften zu Göttingen. Note that the translations which appear in the text are my own, even where published translations are referred to as well as the originals.
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Hallett, M. (2011). Absoluteness and the Skolem Paradox. In: DeVidi, D., Hallett, M., Clarke, P. (eds) Logic, Mathematics, Philosophy, Vintage Enthusiasms. The Western Ontario Series in Philosophy of Science, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0214-1_10
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