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4. Russell: the paradoxes emerge
The paradoxes were not discussed in print until 1903, when they appeared in Russell’s Principles of Mathematics. But Russell had found the paradox of the largest cardinal before he learned of Burali-Forti’s article or of Cantor’s unpublished thoughts on the subject. Russell’s own paradox emerged in May 1901 from simplifying his paradox of the largest cardinal.
But the roots of Russell’s paradox go much deeper. In 1896 he had rejected Cantor’s infinite ordinals as ‘impossible and self-contradictory’. During that period Russell proposed various antinomies in the spirit of Kant and Hegel, including the ‘contradiction of relativity’. In 1898 he characterized mathematics as follows:
One pervading contradiction occurs almost… universally. This is the contradiction of a difference between two terms, without a difference in the conceptions applicable to them. I shall call it the contradiction of relativity. This, with addition and the manifold, appear to define the realm of Mathematics.
By 1899, Russell was no longer an idealist but a Platonist. Although he ceased propounding antinomies in the style of Kant, he proposed one that originated with Leibniz. ‘Mathematical ideas’, he wrote in his 1899–1900 draft of the Principles, ‘are almost all infected with one great contradiction. This is the contradiction of infinity. All antinomies,… so far as they are valid at all, will be found reducible to the antinomy of infinite number’ (1900: 70).
What was this antinomy of infinite number? In 1899 Russell gave it the following form, while distinguishing between
(a) all numbers, (b) the greatest number, (c) the last number. Observe that (b)… means the number applying to the greatest collection. All three are commonly called infinite number, and imply an antinomy, since their being can be both proved and disproved. (a) the most fundamental: There are many numbers, therefore there is a number of numbers. If this be N, N+1 is also a number, therefore there is no number of numbers.
Russell’s antinomy of infinite number has the same formal structure as the paradox of the largest cardinal, which he formulated only in late 1900. That paradox can be stated in the following way: the class of all cardinal numbers has a cardinal number; but there is another class which has a larger cardinal number (by Cantor’s theorem, the set of subsets of a set with cardinal N has a cardinal larger than N); hence there is no cardinal number of the class of all cardinal numbers. Likewise, the antinomy of infinite number has the same formal structure as his later paradox of the largest ordinal.
In November 1900 Russell redrafted part of the Principles. The following passage from that draft manuscript contains the earliest version of his paradox of the largest cardinal, which is closely connected with the antinomy of infinite number.
There is a certain difficulty in regard to the number of numbers, or the number of individuals or of classes. Numbers, individuals, and classes, each form a perfectly definite class, and… every class must have a number. Now the number of individuals must be the absolute maximum of numbers, since every other class is a proper part of this one…. But Cantor has given two proofs… that there is no greatest number.
Soon Russell informed Couturat that Cantor’s proofs were erroneous when applied to the class of all classes. Couturat doubted the existence of that class. But Russell vigorously defended it: ‘If you grant that there is a contradiction in this concept, then the infinite always remains contradictory’ (Moore and Garciadiego 1981: 327).
In May 1901 Russell discovered his paradox by applying Cantor’s proof that there is no greatest cardinal to the class of all classes. During that month he wrote the earliest surviving manuscript containing his paradox. It was not expressed in terms of classes but of predicates:
We saw that some predicates can be predicated of themselves. Consider now those… of which this is not the case…. But there is no predicate which attaches to all of them and to no other terms. For this predicate will either be predicable or not predicable of itself.
Either case implies the opposite, hence a contradiction. Thus there is no predicate predicable of all and only those predicates not predicable of themselves. Russell concluded that the principle of comprehension was false. A year later, he wrote to Frege about it. While Frege saw that the paradox for predicates did not arise in his system, he was devastated by the class form of Russell’s paradox. Frege abandoned the principle of comprehension, which seemed to him the only possible logical foundation for arithmetic.
In 1903 Russell published Russell’s paradox, together with the paradoxes of the largest cardinal and the largest ordinal, in the Principles. By that date, one or more of these three paradoxes was also known to Dedekind and Hilbert (thanks to Cantor), to Husserl (thanks to Zermelo; see §5 below), to Peano(thanks to Russell) and to E.H. Moore (independently, in a letter to Cantor), but none of them published such a result. Russell’s and Frege’s reactions to the paradoxes, which they treated as a major challenge, stand in striking contrast to the early reactions of the others mentioned.
Russell’s book also generalized his paradox: if R is a relation, suppose there is some w such that, for any x, x has the relation R to w if and only if x does not have the relation R to x; substituting w for x leads to a contradiction; hence there is no such w (1903: 102). (Russell’s paradox results from substituting membership for R.) Later Quine expressed Russell’s generalization as follows: nothing can have a relation to all and only those things that do not have it to themselves.
At the end of this book Russell tentatively presented his earliest version of the theory of types as a solution to the paradoxes. Then ‘x is a member of x’ was meaningless, since if x is a member of u, then x and u had to belong to different types (see §7 below).
Moore, Gregory H.. Russell: the paradoxes emerge. 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-the-paradoxes-emerge.
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