The Turing jump

Fix a finite binary syntax for the oracle Turing machines of Oracle Turing machines and an effective decoder of that syntax. Enumerate all binary words, interpreting a malformed word as a program that diverges everywhere; write ${\mathrm{\Phi }}_e^A(x)$ for the partial computation of the $e$-th decoded program on input $x$ with oracle $A$. This is an effective enumeration of every oracle program.

The syntax supports hardwiring: by inserting binary numerals into a fixed program template, there are total computable compiler maps that produce an index for a program with any finite tuple of indices and natural parameters built into it. This is a direct syntactic operation and is uniform in the oracle.

For $A\subseteq \mathbb{N}$, its Turing jump is the relative diagonal-halting set $A^{\mathrm{\prime }}:=\{e:{\mathrm{\Phi }}_e^A(e)\downarrow \}\mathrm{.}$