Example of a recursively enumerable set as a limit of recursive sets I am reading Soare's book and I am trying to understand the following statement:
A function f has r.e. degree iff it is the limit of a recursive sequence ${f_{s}}_{s \in \mathbb{N}}$ and its modulus of convergence $m \leq_{T}f$. 
However, I am trying to find an example to illustrate this theorem. Is it useful to identify the difference between recursive sets and r.e. set?
 A: Sure! The classic example of this is the characteristic function of the Halting Problem, which is recursively enumerable but not recursive.
This function, $f$, is defined as: $f(x)=1$ if $\varphi_x(x)$ halts, and $f(x)=0$ otherwise (where $\{\varphi_e: e\in\mathbb{N}\}$ is some standard enumeration of the partial computable functions).
Now $f$ is a limit of recursive functions in a natural way. Let $f_s(x)=1$ if $\varphi_x(x)$ halts in at most $s$ stages, and $f_s(x)=0$ otherwise. Then each $f_s$ is recursive, and their limit is $f$: $f(x)=1$ iff for all sufficiently large $s$, we have $f_s(x)=1$.  So the quoted result says that $f$ has r.e. degree (and indeed, $f$ itself is r.e.).
So this is an example of a function with r.e. degree being the limit of a sequence of recursive functions. In general, though, such an $f$ need only have $\Delta^0_2$ degree! (This is Shoenfield's Limit Lemma.) And there are lots of $\Delta^0_2$ degrees which are not r.e. degrees. In order to ensure that the limit of a sequence of recursive functions is of r.e. degree, we need to add a technical condition; this is the requirement that the modulus of convergence (that is, at what point $t=m(n)$ we see $f(n)$ "stabilize" - $f_s(n)=f(n)$ for all $s>t$) be "simple". And indeed, it's easy to see that in the example above, the modulus of convergence of $f$ is indeed $\le_Tf$.
