Most of the examples of non-computable real numbers use some kind of a diagonalization construction over some turing computable model of computation. See Are there any examples of non-computable real numbers?.
I want to know if there are "natural" real numbers that are not computable. I'm having difficulty in formalizing what I mean by "natural". Here is a necessary condition for naturality: The description of that number should not mention any turing computable model of computation.
Ideally, this number should have existed in the literature even before Turing invented Turing machines. Somehow this is analogous to the way Solomon Feferman says:
Finally, we must take note of the fact that up to now, no previously (w.r.t. the day Gödel announced his incompleteness theorems) formulated open problem from number theory or finite combinatorics, such as the Goldbach conjecture or the Riemann Hypothesis or the twin prime conjecture or the P=NP problem, is known to be independent of the kinds of formal systems we have been talking about,not even of PA.
in http://math.stanford.edu/~feferman/papers/newaxioms.pdf. The parts in parenthesis have been added by me to put his quote in proper context. My question was partly motivated by this quote.