I am reading a textbook on computability theory and I am struggling to understand creative sets. A theorem presented by the textbook is that suppose that $S \subseteq \varphi_1$ and $A=\{x:\phi_x \in S\}$. If $A$ is recursive enumerable and $A\neq \varnothing$ or $\mathbb{N}$, then $A$ is creative.
However, while I understand the following construction of a simple set and know why it cannot be creative (because it is simple), it seems to me that it conflicts with the above theorem. That is, can we not see Ran($f$) as the index of a subset of unary computable functions? If so, then it is neither empty or equal to $\mathbb{N}$ (I think, as some non-total function $\phi_a$ will diverge before a suitable value of z is found). Furthermore, it is r.e. and non-recursive, so by the above theorem, should it not be creative?
