Turing and enumeration jumps in the Ershov hierarchy

A classification of low c.e. sets and the Ershov hierarchy

“…(cf., e.g., [2, 14]). Moreover, it was proved in [17] that if $$$$\begin{equation*} A^{\prime }\in \Sigma ^{-1}_{\alpha }\setminus {\left(\bigcup _{\beta <\alpha }\Delta ^{-1}_{\beta }\right)} \end{equation*}$$$$for a set A and a computable ordinal $$\alpha >1$$, then α is limit.…”

“…Let S be a system of notation. Recall [17] that a notation $$a\in D_S$$ is normal if $$\nu _S(a)>0$$ and there is a computable function $$$$\begin{equation*} h: \lbrace x\in D_S : \nu _S(x)<\nu _S(a)\rbrace \times \lbrace x\in D_S : \nu _S(x)<\nu _S(a)\rbrace \rightarrow \lbrace x\in D_S : \nu _S(x)<\nu _S(a)\rbrace \end{equation*}$$$$such that the function $$\nu _S\circ h$$ is strictly increasing in both arguments. It is easy to see that each natural notation a of $$\omega ^n$$, $$n>0$$, is normal via the natural sum …”

“…If a set A is jump traceable with $$\Sigma ^{-1}_{\alpha }$$‐bound, then A is jump traceable with $$\Sigma ^{-1}_{\beta +\omega }$$‐bound and, hence, $$A^{\prime }\in \Delta ^{-1}_{\beta +\omega }$$. By [17, Theorem 1] and Theorem 3.1, we have $$A^{\prime }\in \Delta ^{-1}_{\beta }$$. Therefore, A is jump traceable with $$\Delta^{-1}_{\beta}$$‐bound.…”