Turing introduced the fact that the limit of a computable sequence is not necessarily computable, and the Specker sequence is a specific example of such a number (with supremum not computable).
My question is what is known about the Specker sequence in particular and limits of computable reals in general in terms of the type of number it is. The next largest structure being the Arithmetic Hierarchy: are such limits Arithmetic reals?
Furthermore do the Arithmetic reals decompose via the Arithmetic Hierarchy? In other words are there levels: with presumably the Computable reals at level one; level two Arithmetic reals (presumably encoding the Halting function); level three etc, for Arithmetic reals?
If there are such levels do the limits of computable sequence all belong to level 2?