If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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One common method of resolving this is to cash out "representable" numbers in terms of outputs of halting turing machines, so that paradoxes of Berry's sort require solving the halting problem and are therefore not themselves representations.
... Unless there exists something other than a Turing machine that can solve the halting problem.
Which of course leads us to things like "The set of Turing machines that do not halt and cannot be proven to not halt by any Turing machine".