Formal Analogues
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.
George Boolos (1989) built on a formalized version of Berry's paradox to prove Gödel's Incompleteness Theorem in a new and much simpler way. The basic idea of his 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): n has a definition 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 less than k symbols" can be formalized and shown to be a definition in the sense just stated.
Read more about this topic: Berry Paradox
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