I don’t feel safe enough with the vocabulary, so please let me use my own words to describe it. Construcive mathematics implies for me to construct all objects you talk about on the things you already declared (for example, recursion becomes a notational shortcut). So, we need to start with something. One simple approach would be finite bit arrays. The most general approach I see, and the one I take, is to start with computable sequences of bits. Consequently, everything based on that is countable.