Did you mean to say “prove there exists an element of 2^N which is not recursively enumerable” instead, or have some particular constructive system in mind more restrictive than intuitionistic logic? To be explicit, it seems to me the following serves as an intuitionistically acceptable proof that there exists a non-r.e. subset of N:

Let Halt(a, b) mean that the program coded by a halts on input b. Consider {x | ~Halt(x, x)}. If this were r.e. (i.e., semidecidable), then there would be some p such that Halt(p, x) is equivalent to ~Halt(x, x) for all x. But then, Halt(p, p) would be equivalent to ~Halt(p, p), a contradiction (even intuitionistically). Thus, we can conclude, {x | ~Halt(x, x)} is non-r.e.

The only thing which we wouldn’t be able to do intuitionistically, it seems to me, is go on to conclude that this set has a 2-valued characteristic function. I’m almost certain you just made a small typo, but just in case not, have I missed some flaw in the above ostensibly constructive proof?

Perhaps the following exercise will help: prove that there exists a subset of `NN` which is not recursively enumerable. Precisely where did you use classical logic in that proof? Probably it was something like “either `n in W_n` or not.” Now try proving it without this step (you will fail). Welcome to constructive mathematics.

The background needed to understand this, I suspect, is first of all familiarity with classical computability theory, as this is what we’re trying to get at. Then you would also need some familiarity with constructive mathematics.

Your second question, whether a similar theory may be developed using classical logic, is somewhat misguided. If we want to have a synthetic approach to computability based on the Axiom of Enumerability, then it follows that logic is non-classical (see theorem in the notes which states that the Axiom of Enumerability implies that the Law of Excluded Middle is false). I cannot choose logic here, so the answer to your question is “no”, we cannot do the same thing using classical logic. Many people seem to think that mathematics is done in the following order: (1) pick logic (classical or intuitionistic), then (2) do mathematics using the chosen logic. This is not always possible as your choice of mathematical topic may force you to choose a particular logic. For example, you might want to consider an axiom which is inconsistent with classical logic.

I certainly did not ask “How can I do computability theory in intuitionistic logic?” Instead I asked: “The effective topos is a mathematical universe (a model of a certain kind of set theory) which has computability built in. Therefore, it must be ready-made for an abstract treatment of computability theory. What sort of thing does a mathematician who lives in this universe believe in?” (Note: a mathematician who lives in a topos does not see how the topos is constructed. The building blocks of the topos, which to us on the outside look like complicated structures involving Turing machines, look like ordinary sets and functions to him.) As is well known to specialists in this area of reasearch, the answer turnes out to be that mathematicians inside the effective topos happen to believe in the Axiom of Enumerability and happen to disbelieve the Law of Excluded Middle, and that things they say about sets mean to us something about computability.