Consider a Turing machine $T$ with access to an oracle for a proper, nonempty subset of $A_{TM}$, say $L$. That is, $T$ can query this oracle to check whether some string belongs or doesn't belong to $L$. Is it possible to define such an $L$ so that $T$ does not decide $E_{TM}$ (the language of descriptions of Turing machines whose languages are empty)?Intuitively, it seems the problem is that when the oracle answers No, $T$ is unable to distinguish whether the string it fed to the oracle to receive this response belongs to $A_{TM} - L$ or $\overline{A_{TM}}$. But this doesn't really prove that $T$ can't decide $E_{TM}$.
Turing machine with an oracle for a proper subset of a known undecidable language
turing machines;undecidability;oracle machines