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CCC comments on Respond to what they probably meant - Less Wrong
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That's an interesting point. A computer program is itself a finite number of symbols chosen from a finite list (with certain restrictions that further reduce the number of programs that make sense).
The same can be said for the English language. In fact, the same can be said for any language that can be translated into English, or into any other language that has a finite alphabet.
And I don't know how a language with an infinite alphabet would work, but if it can be explained in English, then that implies that it can be translated to English.
This therefore implies that there must exist numbers which cannot be precisely specified at all.
Almost all of them! But, y'know, I can't tell you what any of them are :-).
And among these, there is the smallest undefinable number.
There isn't. The collection of positive, undefinable numbers is bounded below by zero, but doesn't actually have a smallest element (due to the reals not being well-ordered).
I was going to say "Congratulations, you just proved the halting theorem." -- but actually I think the paradox you're gesturing at fails to "work" for shallower reasons (e.g., the reals not being well-ordered -- trivially not by the usual ordering, and less trivially not by anything computable, because it's consistent with ZF for there to be no well-ordering of the reals).
If you mean the smallest undefinable positive number, then isn't that epsilon?