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The 'Berry paradox' is a paradox that arises from self-referential expressions about the smallest possible integer which can be composed in a certain number of words. It was proposed by Bertrand Russell who attributed it to G. G. Berry, a librarian at the Bodleian in Oxford, who had suggested the idea of considering the more limited paradox associated with the expression "the first undefinable ordinal".

Contents

The paradox

Resolution

Sources and notes

See also

Further reading

The paradox

Consider the following expression:
:''The smallest positive integer not definable in under eleven words.''
Since there are finitely many words, there are finitely many phrases of under eleven words, and hence finitely many positive integers that are defined by phrases of under eleven words. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words — that is, positive integers satisfying the property "not definable in under eleven words". By the well ordering principle, if there are positive integers that satisfy a given property, then there is a ''smallest'' positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under eleven words". This is the integer to which the above expression refers; that is, this integer is defined by the above expression. But note that the above expression is only ten words long; so, this integer is defined by an expression that is under eleven words long; so it ''is'' definable in under eleven words, and is ''not'' the smallest positive integer not definable in under eleven words, and is ''not'' defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is, clearly, definable in under eleven words), there cannot be any integer defined by it.

Resolution

The Berry paradox as formulated above arises because of systematic ambiguity in the word "definable." In other formulations of the Berry paradox, such as one that instead reads: "...not nameable in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious-circle fallacies. Other terms with this type of ambiguity are: satisfiable, definable, true, false, function, property, class, relation, cardinal, and ordinal.^[1]
One of the ways it is proposed that this family of paradoxes be resolved is by incorporating stratifications of meaning in language. Terms with systematic ambiguity may be written with subscripts denoting that one level of meaning is considered a higher priority than another in their interpretation. ''The number not nameable₀ in less than eleven syllables'' may be ''nameable₁'' under this scheme. ^[2]
The argument that "''Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words''" assumes that "''there must be an integer defined by this expression''" which is counterfactual as most phrases "''under eleven words''" are ambiguous to their defining of an integer, with this ten word paradox being an example. Assuming one can match word phrases to numbers is a mistaken assumption.^[3]
It is generally accepted that the Berry paradox results from interpreting sets of possibly self-referential expressions: it and similar paradoxes embody so-called "vicious-circle" fallacies. To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them.
Using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, as has been done by Gregory Chaitin. Though the formal analogue does not lead to a logical contradiction, it does prove certain impossibility results, including an incompleteness theorem similar in spirit to Gödel's incompleteness theorem.
Berry's paradox ''can'' be forced into a formal system. George Boolos used a specific formalization to provide an alternate proof of Gödel's Incompleteness Theorem. The basic idea of the proof is that a proposition that holds of ''x'' if ''x'' = ''n'' for some natural number ''n'' can be called a ''definition'' for ''n'', and that the set {(''n'', ''k''): the natural number ''n'' has a definition in this sense that is ''k'' symbols long} can be shown to be representable (using Gödel numbers). Then the proposition "''m'' is the first number not definable in under ''k'' symbols" can be formalized and shown to be a definition in this sense.
The validity of the paradox was challenged in a paper in the ''Journal of Symbolic Logic'', Volume 53, Number 4, Dec. 1988. See ''External links'' below.

Sources and notes

1. Russell/Whitehead, Principia Mathematica
2. Quine, Ways of Paradox
3. In the Journal of Symbolic Logic, Vol. 53, No. 4, 1220-1223. Dec., 1988, in "The False Assumption Underlying Berry's Paradox," by James D. French demonstrated that an infinite number of numbers could be uniquely described in the exact same words.


★ Charles H. Bennett, ''On Random and Hard-to-Describe Numbers'', IBM Report RC7483 (1979)
http://www.research.ibm.com/people/b/bennetc/Onrandom.pdf

★ George Boolos, ''A new proof of the Gödel Incompleteness Theorem.'' ''Notices of the American Mathematical Society'', 36(4), pp. 388-390.

★ Bertrand Russell, ''Les paradoxes de la logique'', Revue de métaphysique et de morale, vol 14, pp 627-650

★ Bertrand Russell and Alfred N. Whitehead, ''Principia Mathematica'', Cambridge University Press/ A paperback reissue up to
★ 56 was published in 1962.

See also

★ Definable number

★ Busy beaver

★ Richard's paradox

Further reading

★ http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm2.html A discussion by Gregory Chaitin

★ http://www.cs.yorku.ca/~peter/Berry.html

★ http://mathworld.wolfram.com/BerryParadox.html The entry for the Berry paradox at Wolfram Research's MathWorld

★ http://www.hgsc.bcm.tmc.edu/~kdurbin/texts/alg.info.chiatin.html