Version: v1, Published online: 1998

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## 2. Burali-Forti: how not to discover a paradox

The paradox of the largest ordinal, often called Burali-Forti’s paradox, is as follows. The set of all ordinals is well-ordered and so has an ordinal number, Ω, which is the largest ordinal. But for any ordinal α, there is a larger ordinal α+1. Hence Ω+1 is larger than Ω, a contradiction.

The early history of this paradox is a comedy of errors. In 1897 Cesare Burali-Forti *almost* discovered it. He failed to do so because he misunderstood Cantor’s definition that a linearly ordered set is ‘well-ordered’ if (a) it contains a first element, (b) every element with a successor has an immediate successor and (c) every set of elements with a successor has an immediate successor. Accidentally omitting (c) from the definition, Burali-Forti introduced a new notion of ‘perfectly ordered class’ and gave for such classes an argument close to the paradox. But the conclusion he drew was that the order types of such classes are not linearly ordered. He did not think he had discovered a paradox, since nothing that he wrote challenged previous results. Later that year, he realized that he had misconstrued Cantor’s definition, but saw no contradiction between his work and Cantor‘s, since his concerned perfectly ordered classes, not ordinals. For five years, there was no discussion in print about Burali-Forti’s article. (This has often been mistakenly denied, as when van Heijenoort stated that Burali-Forti’s article ‘immediately aroused the interest of the mathematical world’ (1967: 104).)

In 1902, after Louis Couturat alerted him to Burali-Forti’s work, Russell asserted (wrongly) in print that Burali-Forti claimed that the ordinals are not linearly ordered. Here was the beginning of a genuine paradox, for Russell applied Burali-Forti’s argument to the ordinals. Russell’s solution was to claim that the class of all ordinals has no ordinal since it is not well-ordered. (Later he rejected this solution.)

In 1903 Russell gave the first clear statement of the paradox of the largest ordinal in his book The Principles of Mathematics. Thus Russell was the key figure in its emergence. Alonzo Church erroneously claimed that ‘it was through Burali-Forti’s paper that there first came to general attention the threat to the foundations of mathematics that is constituted by the antinomies’ (1971: 235). It was not Burali-Forti but Russell who first pointed out this threat in print.

Moore, Gregory H.. Burali-Forti: how not to discover a paradox. 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/burali-forti-how-not-to-discover-a-paradox.

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