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Sniffnoy comments on The Pascal's Wager Fallacy Fallacy - Less Wrong
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I'm not sure what a "nameable number" is. Whatever countable naming scheme you invent, I can "name" a number that's outside it by the usual diagonal trick: it differs from your first nameable number in the first digit, and so on. (Note this doesn't require choice, the procedure may be deterministic.) Switching from reals to nameable numbers seems to require adding more complexity than I'm comfortable with. Also, I enjoy having a notion of Martin-Löf random sequences and random reals, which doesn't play nice with nameability.
By "nameable number" he seems to just mean a definable number - in general an object is called "definable" if there is some first-order property that it and only it satisfies. (Obviously, this dependson just what the surrouding theory is. Sounds like he means "definable in ZFC".) The set of all definable objects is countable, for obvious reasons.
With this definition, your diagonal trick actually doesn't work (which is good, because otherwise we'd have a paradox): Definability isn't a notion expressible in the theory itself, only in the metatheory. Hence if you attempt to "define" something in terms of the set of all definable numbers, the result is only, uh, "metadefinable". (I gave myself a real headache once over the idea of the "first undefinable ordinal"; thanks to JoshuaZ for pointing out to me why this isn't a paradox.)
EDIT: I should point out, using definable numbers seems kind of awful, because they're defined (sorry, metadefined :P ) in terms of logic-stuff that depends on the surrounding theory. Computable numbers, though more restrictive, might behave a little better, I expect...
EDIT Apr 30: Oops! Obviously definability depends only on the ambient language, not the actual ambient theory... that makes it rather less awful than I suggested.
We could similarly argue that the definable objects should be thought of as "meta-countable" rather than countable, right? The reals-implied-by-a-theory would always be uncountable-in-the-theory. (I'm tempted to imagine a world in which this ended the argument between constructivists and classicists, but realistically, one side or the other would end up feeling uneasy about such a compromise... more likely, both.)
I think you're confusing levels here. When I spoke of "the surrounding theory" above, I didn't mean the, uh, actual ambient theory. (Sorry about that -- I may have gotten a little mixed up myself) And indeed, like I said, definability only depends on the language, not the theory. Well -- of course it still depends on the actual ambient theory. But working internal to that (which I was doing), it only depends on the language. And then one can talk about the metalanguage, staying internal to the same ambient theory, etc... (mind you, all this is assuming that the ambient theory is powerful enough to talk about this sort of thing).
So at no point was I intending to vary the actual ambient theory, like you seem to be talking about.
Warning: I don't quite understand just how logicians think of these things and so may be confused myself.