A critique of "Berry's paradox". Is it even necessary?
To be written in the future.
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
DiscussionA 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 syllablesis corresponded with, because no such rules of correspondence are provided. I rest my case.