[lutusp]

> Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions.

I'm tempted to say that that definition places those examples in a unique set, thus at least unambiguously identifying the set to which they belong.

[mtdewcmu]

This appears to be exactly the concept I was talking about. Maybe I read about these numbers some time ago and forgot that they already had a name.

http://en.wikipedia.org/wiki/Definable_number

[mtdewcmu]

You can identify the set of numbers with infinite Kolmogorov complexity. But you can't separate out an individual from the set.

Turing machines might not capture all numbers that can be described, but, interestingly, descriptions and Turing machines have the same cardinality.