[mtdewcmu]

"Many irrational numbers are easy to identify unambiguously."

Absolutely. Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions. Some numbers can only be described by an infinitely long list of digits, one that can't be produced by a Turing machine and contains an infinite amount of irreducible information. The Kolmogorov complexity is infinite.

Newton's equations describe a mathematical model of the universe that agrees well with measurements taken under familiar conditions. Einstein's equations describe a different mathematical model, one that has good agreement with experiment over a much wider range of conditions than Newton's. But general relativity has well-known problems. Its equations give nonsense solutions under some circumstances, e.g. singularities. Physical theories are models that predict the outcomes of experiments. They're reductionist out of necessity. It's unknown and probably unknowable as to whether a perfect model is possible, but there are likely things that can't be reduced.

[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.