A set $B$ is $\Delta_{n+1}$ if and only if $B \leq_T \emptyset^{(n)}$. More generally, $B$ is $\Delta^C_{n+1}$ if and only if $B \leq_T C^{(n)}$.
This is from http://en.wikipedia.org/wiki/Post%27s_theorem and my question is, what exactly is $\Delta_{n+1}$ representing? Is it $\Delta_{n+1}^0$ in ordinary arithmetic hierarchy notation?
Also, what exactly is $\Delta^C_{n+1}$ here? Is it $\Delta^{0,C}_{n+1}$ in ordinary relativization oracle notation?
On
Here is another characterization of $\Delta^0_n$. A subset of the natural numbers is $\Delta^0_n$ if and only if both it and its complement are $\Sigma^0_n$. A subset $X$ of the natural numbers is $\Sigma^0_n$ if and only if $X$ can be defined as follows, $k\in X$ if and only if $\exists m_1\forall m_2\exists m_3\dots Q_n m_n R(k,m_1,\dots m_n)$, where $R$ is computable and $Q_n$ is $\exists$ if $n$ is odd and $Q_n$ is $\forall$ if $n$ is even. For example, when $n$ is equal to 1, $X$ is $\Sigma^0_1$ if and only if it has an existential definition if and only if it is computably enumerable. It is $\Delta^0_1$ if and only if both it and its complement are computably enumerable if and only if it is computable. Shoenfield's theorem is a natural extension of that example.
Yes, the superscript $C$ indicates relativization to $C$ as oracle. Equivalently, one can think that $C$ is given as data.
Yes, the article means to say $\Delta^0_{n+1}$ and $\Delta^{0,C}_{n+1}$.