I see what you're saying, and yeah, I think you're right. You can construct a non-computable function S(n) that returns n when n is even, but does something non-computable when n is odd (kind of like f_bar). Of course, we're not really interested in these degenerate cases where n is even, but rather in the cases where we know that there's some kind of answer we can't prove in the system. I gave "S(1) = 0" as an example (you can replace '1' with whatever argument ends up being non-computable in your case, maybe S(2)).