Version: v1, Published online: 1998

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## 6. Berry, König and Richard: paradoxes of definability

The first paradoxes of definability emerged during 1904–5 from three distinct sources. Richard’s paradox, due to Jules Richard, was the first to be published. Conceived as a response to Hadamard’s comments on the paradox of the largest ordinal (see §5 above), but directed at the continuum of real numbers, Richard’s paradox was the following: consider in decimal form all the real numbers that can be defined by a finite number of words. These numbers form a denumerable set *E*, whose members are a_{1},a_{2},a_{3},…. We define a real number *N* not in *E* by letting the *n*th decimal place of *N* be one more than the *n*th place of a_{n} unless the *n*th place is 8 or 9 (in which case the *n*th place of *N* is 1). Since *N* differs from the *n*th place of a_{n} for each *n*, then *N* is not in *E*. But *N* is defined by a finite number of words and so is in *E*, a contradiction.

In 1906 Henri Poincaré stressed the importance of Richard’s paradox, and found in it a single explanation for all the paradoxes: objects defined by a vicious circle must be avoided. In Richard’s paradox, there is a vicious circle since *N* is defined by using *E* but is itself in *E*. If one restricted *E* to those numbers defined without reference to *E*, there would be no contradiction. Yet Poincaré viewed the actual infinite, which he rejected, as being equally the source of the paradoxes.

Although it was the first definability paradox to be published, Richard’s was not the first to be discovered. That honour goes to George Berry of Oxford University. In December 1904 Berry sent Russell the following paradox: consider the least ordinal not definable in a finite number of words; this ordinal is itself defined in a finite number of words, a contradiction.

This paradox, which Russell published in 1906 in response to Poincaré’s comments on Richard’s paradox, is known by the name he gave it: the paradox of the least indefinable ordinal. At that time Russell modified this paradox to give one that did not involve the actual infinite and that he called Berry’s paradox, although Berry did not formulate it: the least integer not nameable in fewer than nineteen syllables. But this integer has just been named in eighteen syllables, a contradiction.

A fourth paradox of definability, called the Zermelo–König paradox, was found in 1905 by Julius König, who published it not as a paradox but to refute Zermelo’s claim that the continuum of real numbers can be well-ordered. König observed that the set of real numbers which are definable in a finite number of words is denumerable. Since the continuum is not denumerable, there are real numbers *x* which cannot be defined in a finite number of words. If the continuum can be well-ordered, then there is a least such *x*, a contradiction. Hence the continuum cannot be well-ordered.

Once Richard’s paradox was seen to be similar to König’s result, it became clear that they concerned definability and had nothing to do with well-ordering the continuum.

Moore, Gregory H.. Berry, König and Richard: paradoxes of definability. 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/berry-konig-and-richard-paradoxes-of-definability.

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