numbering turing programs

since each turing program is a finite set of quintuples, we can list all turing programs in such a way that for any program we can effectively find its number on the list, and conversely. using godel numbering.

let $\text{[math]}$ be the turing program with code number $\text{[math]}$ in this coding, also called the godel number or index $\text{[math]}$ .
let $\text{[math]}$ be the partial function of $\text{[math]}$ variables computed by $\text{[math]}$ . let $\text{[math]}$ denote $\text{[math]}$ . we call $\text{[math]}$ a partial computable function. we identify the turing program $\text{[math]}$ and p.c. function $\text{[math]}$ with index $\text{[math]}$ .

[cite:;taken from @computability_soare_2016 definition 1.5.1]

this is called the standard numbering or canonical numbering of turing program and partial computable functions. any other effective numbering will be computably isomorphic to this one.
the second method uses the standard pairing function and has the added advantage that the $\text{[math]}$ -tuple coding is an injective and surjective map from $\text{[math]}$ onto $\text{[math]}$ .
[cite:;taken from @computability_soare_2016 chapter 1 defining computability]