On Random and Hard-to-Describe Numbers

February 19, 2021 Jeremiah Hendren

On Random and Hard-to-Describe Numbers

Bennett, Charles H.
In Randomness And Complexity, From Leibniz To Chaitin, pp. 3-12. 2007
https://doi.org/10.1142/9789812770837_0001

The number 1,101,121 is unusual in that it is, or appears to be, the number named by the expression 'the 1st number not nameable in under ten words.' However, since the italicized expression has only nine words, there is an inconsistency in regarding it as a name for 1,101,121 or any other number. This paradox, a variant of one due to Russell and Berry, shows that the concept of nameability or definability is too vague and powerful to be used without restriction. Because of it, the "function" N(x) = 'the number of English words required to name the integer x' must be regarded as ill-defined for all but finitely many x. Martin Gardner has pointed out that a similar paradox arises when one attempts to classify numbers as "interesting" or "dull": there can be no dull numbers, because, if there were, the 1st of them would be interesting on that account.

Charles H. Bennett

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