A critique of "Berry's paradox". Is it even necessary?

Last modified: 
21/3/2017
To be written in the future.

Berry's paradox

There are several versions of the Berry paradox, the original version of which was published by Bertrand Russell and attributed to Oxford University librarian Mr. G. Berry. In the form stated by Russell (1908), the paradox notes that, "'The least integer not nameable in fewer than nineteen syllables' is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction."1

Discussion

A basic practice, when writing mathematical statements, is to define one or more rules, which will serve as a framework to unambiguously enough define correct notation and well-formulated or valid statements.

The problem with Berry's paradox, is that no such framework is provided. That is, it is not at all clear which rules allow syllables to be named by, or be corresponded with, integers. After all, there is not even a fixed syllable set, considering continuums such as tone, nasalization, breathiness, vowel space, aspiration. So, eightteen syllables really suffice to create an infinite number of integers, if you would like.

In fact, even if one would be limited to one syllable, it is still unclear how to proceed with assigning numbers to that item of pronunciation, which can of course be pronounced in infinitely many ways. And even if it would only be able to be pronounced in one way, one could still assign an arbitrary amount of numbers to that item.

To be more succinct, as one create functions which relate all integers to one syllable, it is still not at all clear which integer the string The least integer not nameable in fewer than nineteen syllables is corresponded with, because no such rules of correspondence are provided. I rest my case.

P.S.

Another way to attack this paradox is by stating that one tries to define something in two contradictory terms. That's hardly a paradox. It's analogue to stating that a red circle defines the number 1, and then later on stating that the same red circle defines number 2.

Footnotes

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