Sorry about drunkenly misremembering things tonight. I was after half-remembered Chaitin's Constant (http://en.wikipedia.org/wiki/Chaitin%27s_constant), which is definable but incomputable.
Hmm, yes. That differs from the construction we discussed last night (a binary expansion with bit i set if the program represented by i halts), but both have the property of being a non-computable real which you can precisely mathematically define.
Chaitin's constant contains less information than the silly binary expansion number, but since it's a probability it's almost justifiable as something you might actually want to talk about, unlike the binary expansion number which is very obviously only a pathological counterexample :-)
It's one I've heard before, though. It's probably the single most natural thing to spring to your mind if someone challenges you to define an uncomputable real. Chaitin's version is more elegant but less obvious.
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Chaitin's constant contains less information than the silly binary expansion number, but since it's a probability it's almost justifiable as something you might actually want to talk about, unlike the binary expansion number which is very obviously only a pathological counterexample :-)
The one I mentioned last night is, as far as I can tell, entirely my own bitrotted rubbish!