Recall Kleene-Post's theorem says that there exists A and B $\leq_T \emptyset'$ that are incomparable.
Recall $\cup_s \sigma_s= A$ and $\cup_s \tau_s= B$ where $\sigma_s$ and $\tau_s$ are decided with oracle $\emptyset'$.
Why is it that given an oracle $\emptyset'$, A and B become decidable? I think the reason is that given x $\in \omega$ we may decide its existence in A by computing $\sigma_x$ with each $\sigma_{y \leq x}$ taking finite construction time.
$A\le_T \emptyset^\prime$ means we can find a $\emptyset^\prime$-recursive function of a characteristic function of $A$. That is, under the oracle $\emptyset^\prime$, we have a effective procedure that computes the characteristic function of $A$.