Version: v1, Published online: 1998
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7. Russell: solving the paradoxes by type theory
In 1905 Russell discussed three kinds of solutions to the paradoxes: zigzag theories, theories of limitation of size, and no-classes theories. In a zigzag theory, a property φ(x) determines the class of all x satisfying φ(x) when φ(x) is sufficiently simple; the difficulty with such a theory, Russell noted, was in specifying what made φ(x) sufficiently simple. A theory of limitation of size, such as that of Cantor or later Zermelo, would state that a property φ(x) determines the class of all x satisfying φ(x) if the class is ‘limited’ in size; the difficulty here was to specify how large an ordinal exists. Finally, a no-classes theory would dispense with assuming the existence of classes and operate directly with properties; although such a theory would preserve classical analysis and geometry, it was unclear how much set theory would survive.
Russell also gave a general form to all paradoxes about classes: suppose that there is a property φ and a function f such that if φ is satisfied by all members of the class u, then f(u) satisfies φ and is not a member of u; the assumption that there is a class w of all x satisfying φ and that f(w) exists leads to the contradiction that f(w) has and does not have the property φ. In Russell’s paradox, φ is ‘x is not a member of x’ and f(u) is u; in the paradox of the largest ordinal, φ is ‘x is an ordinal’ and f(u) is the ordinal number of u.
In the Principles (1903) Russell had tentatively presented his earliest version of the theory of types as a solution to the paradoxes. But during 1903 Russell became dissatisfied with his theory of types, and in 1904 adopted a zigzag theory. In 1905 and 1906 he preferred a no-classes theory. But the boundaries between his different theories were fuzzy. In 1906 he stressed that his no-classes theory contained types and was close to his 1902 theory of types.
The emergence of definability paradoxes coincided with a shift in Russell’s views. Earlier, he had primarily investigated paradoxes involving classes. In 1906 he was increasingly occupied with the liar paradox. In that context he formulated the vicious circle principle, which any proposed solution to the paradoxes must satisfy: whatever involves a bound variable must not be a possible value of that variable. While accepting Poincaré’s analysis that vicious circles caused the paradoxes, Russell vigorously rejected Poincaré’s claim that the actual infinite caused them too.
In 1908 Russell published his ramified theory of types, which included much of the no-classes theory. Whereas in 1900 Russell had vigorously defended the class of all classes, his theory of types explicitly rejected it (as well as the notions of all propositions, all relations, all definitions and all ordinals). The heart of the theory was the distinction between orders and types, where orders concerned definitions, and the central assumption was his dubious axiom of reducibility, which was used to solve the semantic paradoxes.
Russell’s theory was not well received at first, and was rejected by Poincaré in part because of the axiom of reducibility. The simple theory of types, which dispensed with this axiom and with definability, was due to Leon Chwistek and Ramsey in the 1920s (see Theory of types). In the simple theory, there was no concern with the semantic paradoxes, which could not even be formulated in it.
Moore, Gregory H.. Russell: solving the paradoxes by type theory. Paradoxes of set and property, 1998, doi:10.4324/9780415249126-Y024-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/paradoxes-of-set-and-property/v-1/sections/russell-solving-the-paradoxes-by-type-theory.
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