From Ullman's Introduction to Automata Theory, Languages and Computation, in a TM with oracle $A$ :
Observe that if $A$ is a recursive set, then the oracle $A$ can be simulated by another Turing machine, and the set accepted by the TM with oracle $A$ is recursively enumerable.
On the other hand, if $A$ is not a recursive set and an oracle is available to supply the correct answer, then the TM with oracle $A$ may accept a set that is not recursively enumerable.
Does the second sentence mean that a TM with oracle being r.e. accepts a non r.e. language?
Is it correct that there is no automaton which may accept a non r.e. language?
So does a TM with oracle being r.e. not exist?
Is a TM with oracle also a TM?
Is a TM with oracle being r.e. not a TM? (I guess so, because the language accepted by a TM must be r.e.)
For example, I am trying to understand the difference between a many-one reduction and a Turing reduction.
A many-one reduction from language $L1$ to another $L2$ is defined as an algorithm i.e. a Turing machine which ....
A Turing reduction from language $L1$ to another $L2$ is defined as a Turing machine with oracle $L2$ and accepting $L1$. Is a Turing reduction from a language to another necessarily an algorithm i.e. a TM?