Paradoxes of set and property

DOI: 10.4324/9780415249126-Y024-1
Version: v1,  Published online: 1998
Retrieved December 05, 2020, from

5. Hilbert, Bernstein and Grelling: German reaction to the paradoxes

Cantor was not alone in recognizing, before paradoxes were published, that there is no set of all sets. In his 1890 book on the algebra of logic, Ernst Schröder rejected such a universal set, arguing that it leads to a contradiction. Reviewing Schröder’s book, Edmund Husserl rejected both Schröder’s argument and his conclusion, since both resulted from a failure to distinguish between being a member of a set and being a subset. In 1902 Ernst Zermelo informed Husserl that Schröder was correct in his conclusion, although mistaken in his argument. Zermelo showed that a set which contains each of its subsets as a member is self-contradictory. He did so by stating Russell’s paradox, which he had found by 1900, before Russell. Zermelo concluded that there is no set of all sets.

In 1903 Hilbert wrote to Frege that Zermelo had discovered Russell’s paradox three years earlier, after Hilbert had informed Zermelo of other paradoxes which Hilbert had found a year before that. (In an 1899 letter to Frege, Hilbert had observed that there is no set of all cardinal numbers, but gave no hint that this was a threatening discovery.) Influenced by Frege rather than Russell, Hilbert first published on the paradoxes in 1905. He rejected the principle of comprehension, regarding it as the source of the paradoxes. These showed that traditional logic did not satisfy the demands of set theory; they would be solved only by an axiomatic development of logic. That year he devoted a long series of lectures to the problem without coming to a solution. Although one often reads that he did not concern himself with logic during the next decade, in fact he repeatedly lectured on the paradoxes. He only reached a solution in 1917 when he adopted Russell’s theory of types (see §7 below).

In Germany the first published discussion about the paradox of the largest ordinal occurred in 1905. Felix Bernstein used this paradox while attempting to refute Zermelo’s proof (1904) that every set can be well-ordered, arguing that there is indeed a set W of all ordinals (Moore 1982: chap. 2). But in France, Jacques Hadamard rejected Bernstein’s argument: ‘It is the very existence of the set W that generates a contradiction…. One has the right to form a set only with previously existing objects, and it is easily seen that the definition of W presupposes the opposite’ (Hadamard in Moore and Garciadiego 1981: 339).

Several neo-Friesian philosophers commented on the paradoxes, including Alexander Rüstow (1910), in his book on the liar paradox, and Gerhard Hessenberg (1906), who found the notion of set unclear. Hessenberg argued that Russell’s paradox was not dangerous for mathematicians, since they did not use a universal set, but that the paradox of the largest ordinal was dangerous.

In 1908 the neo-Friesian philosophers Kurt Grelling and Leonard Nelson discussed those paradoxes, including the predicate form of Russell’s paradox. Grelling formulated a related paradox, later called Grelling’s paradox: a word is called autological if it applies to itself, and otherwise is called heterological. Thus ‘short’ is autological and ‘long’ is heterological. Now ‘heterological’ is either autological or heterological. But either possibility leads to the opposite.

Citing this article:
Moore, Gregory H.. Hilbert, Bernstein and Grelling: German reaction to the paradoxes. Paradoxes of set and property, 1998, doi:10.4324/9780415249126-Y024-1. Routledge Encyclopedia of Philosophy, Taylor and Francis,
Copyright © 1998-2020 Routledge.

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