This isn't a paradox. It's simply a proof that all natural numbers are interesting, if we accept the premises. We shouldn't accept the premises, because there

This isn't a paradox. It's simply a proof that all natural numbers are interesting, if we accept the premises. We shouldn't accept the premises, because there really is a real paradox lurking around the corner, but this isn't it. The paradox occurs when we replace "natural number" by "ordinal". Then the same line of reasoning demonstrates that all ordinals must be interesting. But surely there can only be countably many interesting ordinals! (Because "interesting" is really code for "definable".) Which shows what the real problem in the premises were -- we failed to distinguish between definable in some particular theory and definability in the meta-theory. See: http://en.wikipedia.org/wiki/Berry_paradox http://en.wikipedia.org/wiki/Berry_paradox