I asked myself similar questions before, for example "Are the definable real numbers countable"? It seemed to me that the set of all explicitly and unambiguously definable objects is "countable", because we could write down the text defining a specific object in the English language. (Or use the English language to define a formal language appropriate to write down the definition.) But English isn't unambiguous, and the meaning of "countable" depends on the used set theory.
Right now I'm thinking about second order logic. Assuming that the natural numbers exist and are well defined seems unproblematic to me. However, the question which subsets of the natural numbers (should we assume to) exist has no good answer. If we say that "all" subsets exist, we run into all sorts of set theoretic questions. If we resort to axiomatic set theory for these questions, we end up with the fact that the natural numbers can't be defined up to isomorphism. The alternative seems to be to assume that only the definable subsets of the natural numbers exist. But my guess is that this won't work out either, i.e. that the "set of all definable subsets of the natural numbers" turns out to be a very strange beast. But if it is a strange beast, I guess it won't be recursively enumerable.
Question Assuming the Church-Turing thesis (or a suitable analogue for definability, or some other sufficiently solid foundation), what can be said about the "set of all definable subsets of the natural numbers"? Is it well defined and recursively enumerable?
Edit It has been pointed out that my use of "recursively enumerable" is ambiguous without further clarifications, especially that I have to specify how a definable subset of the natural numbers is expected to be encoded as a natural number. Given a natural number, take its binary representation and interpret it as a document in the "OpenDocument" format (ODF). Assume that we can efficiently decide whether the document is valid ODF and whether its content is written sufficiently clear and tries to define a subset of the natural numbers. Let's call this a well formed definition. Assume further that each definable subset has at least one well formed definition which can be recognized to succeed in defining a subset of the natural numbers. Now the question is whether there exists a recursively enumerable subset $S$ of the natural numbers such that each number in $S$ corresponds to a well formed definition which succeeds, and that for each definable subset the set $S$ contains a corresponding definition.