The Turing degrees $D$ is defined to study those properties of the subsets of the natural numbers $\mathbb{N}$ which can be expressed in terms of relative computability. Formally, suppose $A$ and $B$ are subsets of $\mathbb{N}$, we say $A$ is Turing reducible to $B$ and write $A\leq_{T} B$ if there is a computational procedure which takes an input $n$ from $\mathbb{N}$, asks whether various numbers are in $B$, and, after finite computational steps if it receives correct responses to those questions returns the answer to whether $n$ is an element of $A$. Indeed, by a given $B$, we would be able to compute $A$. The Turing degrees will be the classes of the equivalence relation $\equiv_{T}$ arising of $\leq_{T}$. An interesting thing is that $D$ with Turing reducibility relation ( $D, \leq_{T} $ ) is a partial ordering structure namely degree structure. I like to take a general view of the degree structure. My questions in that
- what effective data is gained from studying the degree structure on $\mathbb{N}$ in computability theory?
- Is this a unique way to define a degree structure on $\mathbb{N} $?
- What potential difficulty is there to generalize the definition of the degree structure on $\mathbb{N}$ to a degree structure on $\mathbb{Z}$?