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Can't the set of effectively representable numbers be inductive if we decide that "the smallest number not effectively representable" does not effectively represent a number?
"The smallest positive integer not definable in under twelve words" isn't an effective representation of a number any more than "The number I'm thinking of" or "Potato potato potato potato potato" are.
Sure, that's exactly what we have to do, on pain of inconsistency. We have to disallow representation schemas powerful enough to internalise the Berry paradox, so that "the smallest number not definable in less than 11 words" is not a valid representation. Cf. the various set theories, where we disallow comprehension schemas strong enough to internalise Russell's paradox, so that "the set of all sets that don't contain themselves" is not a valid comprehension.
Nelson thought that, similarly to how we reject "the smallest number not... (read more)