Berrys Paradox

[From AreTheSpecifiableRealsWellDefined]

To clarify, the situation is a form of Berry's Paradox.

For the purposes of the original form of this paradox only (given below), definitions may be taken to be formed from words drawn from a finite dictionary (so that only finitely many definitions exist formed from a given number of syllables).

Consider the phrase, 'The least positive integer not definable in fewer than twenty-four syllables.'. If one assumes it is a valid definition, one gets an immediate contradiction, since it has only twenty-three syllables. Hence, it is not a valid definition. (This is known as Berry's "paradox".)

This "definition" involves a form of self-reference - it implicitly refers to every definition (using fewer than twenty-four syllables) of a positive integer, but itself (if valid) is such a definition.

Clearly, one has to disallow such self-reference, but it's awkward to be precise as to what is disallowed without finding that the set of positive integers requiring at least twenty-four syllables for definition exists but can't be referred to in any (valid) definition!

See http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm2.html for a transscript of a lecture by GregoryChaitin on the subject of the Berry paradox (and how he tried to meet with KurtGoedel but never succeeded).

CategoryMath CategoryLogic