Abstract
It is a great pleasure to contribute to this Festschrift for John Bell. No-one has done more than he has to demonstrate the fruitfulness of the interplay between technical mathematics and philosophical issues, and he is an inspiration to all of us who work somewhere in the borderland between mathematics and philosophy.
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Notes
- 1.
Logical cognoscenti know that this is false, but the view is common amongst those who are mathematically, but not logically, well-informed.
- 2.
As far as I know, nobody has ever explicitly put forward the naive conception, though it is implicit in (Frege, 1893).
- 3.
For the definition of well-founded, see below.
- 4.
For more details, including attempts to apply the same idea to the semantic paradoxes, see (Rieger, 1996, Ch. 1) or (Priest, 1994) (Priest uses the terms transcendence and closure). Dummett’s idea of an indefinitely extensible concept (Dummett, 1991, p. 316) and the book (Grim, 1991) are also relevant. The basic idea can be found in (Russell, 1906c), which I discuss below.
- 5.
Though it is not quite as explicit, the distinction between the transfinite and the absolute infinite can be found much earlier in Cantor’s writings (e.g., Cantor, 1883, p. 205). It seems likely that, having realized that any set can be enlarged by the power set operation, Cantor drew immediately the conclusion that there can be no universal set. Cantor is sometimes accused of believing in naive set theory (e.g., (Körner, 1960, p. 44): “Cantor’s theory of classes, by admitting as a class any collection, however formed, leads to contradictions”). This is quite unjustified: rather “his conception of set … was one in which the paradoxes cannot arise” (Menzel, 1984, p. 92); see also (Hallett, 1984, p. 38 and passim).
- 6.
More precisely, the principle that a collection is too big to form a set iff it can be put into 1-1 correspondence with the universe can be taken as the basis for an axiomatization of set theory, as is done in (von Neumann, 1925).
- 7.
A set x satisfying this condition is said to be well-founded.
- 8.
To make the example work, interpret “discussed in this paper” so that it applies to only a small (e.g. finite) number of things.
- 9.
- 10.
“Term” here just means “object.”
- 11.
It might be thought that “limitation of size” embodies exactly the idea of restricting universality, but it is clear that Russell does not think of it in this way: rather he sees the theory as posing the question “how far up the series of ordinals it is legitimate to go” (p. 53), a question which he cannot see any prospect of answering.
- 12.
The original has “of,” which seems to be a misprint.
- 13.
The occurrence of “any” (and “some”) may seem puzzling here: since anything presumably concerns itself, the principle seems to rule out anything ever being a member of a class. But Russell should be read as forbidding any member of a class concerning quantification over the class.
- 14.
To avoid confusion, I should perhaps make it clear that here and throughout the paper I use “constructivism” as a name for a metaphysical view about mathematics, roughly that mathematical objects are brought into existence by some activity of human minds. The term is sometimes now used for mathematics without the law of excluded middle, but I shall use it in its earlier sense.
- 15.
This is a semantic paradox, introduced in (Richard, 1905), which concerns the collection E of all reals definable in a finite number of words; by a diagonal argument we can obtain a new real, not in E yet definable in a finite number of words.
- 16.
The original italicises this sentence. At this point “non-predicative” means simply “not defining a class”; confusingly, Russell, having accepted the diagnosis, started using “impredicative” to mean “violating the vicious circle principle.”
- 17.
My italics.
- 18.
Goldfarb (1989) attempts to reconcile Russell’s predicativism with his lack of constructivism, arguing that his views on variables and their ranges of significance can lead to ramification of intensional entities (in particular propositions and propositional functions) even on a realist conception. But even if this is right—and Goldfarb says he is only making a “first step” (p. 27) towards a full treatment of the issue—the fact remains that Russell advocates the VCP in full generality. As Goldfarb admits (pp. 30–31), it is hard to see how the ramification of sets can be justified except on a constructivist view.
- 19.
The axiom of foundation immediately rules out a universal set, for such a set would be a member of itself. But the point is that such a set is ruled out anyway by the other axioms. Foundation plays no role in solving the paradoxes.
- 20.
- 21.
Gödel claims to discern a third principle, concerning “pre-supposing,” which I shall not discuss here.
- 22.
This will not do as it stands as a characterization of impredicative definitions. For example it will be equally objectionable if, instead of x itself being a member of the totality, some second object y, defined using x, is a member. Presumably to make this rigorous we would require some notion of well-foundedness for definitions; I shall not attempt to supply details here.
- 23.
The transitive closure of x is the set whose members are the members of x, the members of the members of x, and so on.
- 24.
Though he does not state it explicitly, this seems to be what Gödel has in mind in his paper (see p. 131 with its footnote reference to Mirimanoff). It is a little too strong to say that VCP II entails the axiom of foundation, for an infinite descending membership chain \(x_{1} \ni x_{2} \ni \dots \) in which all the x i are different violates foundation without circularity. Such a chain seems equally offensive to the constructivist intuitions underpinning the VCPs, and suggests that they do not fully capture those intuitions.
- 25.
There is disagreement, for example, on whether the axiom of replacement is derivable from the iterative conception.
- 26.
A variation is possible in which instead we start with some atoms or urelements, that is, some non-sets. Though this is probably more natural from a naive point of view, mathematicians standardly work with a universe of pure sets, where everything is a set, since this is technically smoother (for example the quantifiers can simply be taken to range over all sets) and does not result in any limitation in structure. For present purposes the difference in the two approaches is not important.
- 27.
More detailed marshallings of evidence against the iterative conception may be found in (Lavine, 1994, Ch. V; Hallett, 1984, Chs. 5–6). The overall conclusion of Hallett’s book, however, that “we have no satisfactory simple heuristic explanation of why it [ZF] works,” seems to me to be too strong. It is not mysterious that ZF avoids the paradoxes, since it is apparent from the axioms that the paradoxical collections are denied sethood. Hallett also makes much (in Chapter 5) of the technical result that we have very little idea of the size of the power set of ω, arguing that this refutes ZF’s claim to embody a “limitation of size” conception. This, however, seems to depend on thinking of “limitation of size” in the style of Russell, as “no sets allowed that are bigger than such-and-such a cardinal”; rather, as I have been trying to convey, the point is that however big it is, P(ω) is still a set, and therefore not as large as the universe. There is, however, another sense of “why ZF works” considered by Hallett: why it (or indeed any set theory) is adequate as a foundation for mathematics. I agree that this is genuinely mysterious, and I shall not try to solve the mystery here.
- 28.
Boolos is using “ Sx” for “x is a set”.
- 29.
- 30.
I am simplifying the details of the theory to bring out the essential point.
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Rieger, A. (2011). Paradox, ZF, and the Axiom of Foundation. 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_9
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