TITLE: Category of Gödel Codings? [Reference Request]
QUESTION [8 upvotes]: Consider computation with the integers $\mathbb{Q}$.  The traditional theory of recursive functions on $\mathbb{N}$ applies to $\mathbb{Q}$ by the identification of $\frac{a}{b} \in \mathbb{Q}$ with $2^{a}3^{b} \in \mathbb{N}$.  Similarly, if I have any similar unnamed structure I can proceed likewise, as follows:
First, note that in this process of "Gödel numbering" a given object may have more than one representative Gödel number in $\mathbb{N}$.  However the set of valid Gödel numbers is primitive recursive and there is a primitive recursive equivalence relation that determines whether two numbers represent the same object.
This leads to the following category $\textsf{PRA}$:  The objects of $\textsf{PRA}$ objects are pairs $(A, \sim _{A})$ where $A \subseteq \mathbb{N}$ is a primitive recursive set and $\sim _{A} \subseteq A \times A$ is a primitive recursive equivalence relation.  The morphisms of $\textsf{PRA}$ are are functions $f: (A, \sim _A) \rightarrow (B, \sim _B)$ induced by primitive recursive functions $F: \mathbb{N} \rightarrow \mathbb{N}$ such that $x \sim _{A} y \Rightarrow F(x) \sim _{B} F(y)$.
This is all rather rudimentary and seems like the correct definition (to me) of a Gödel numbering and leads naturally to the category just described.  However I have never seen this category described or studied.  Is this category simply

Uninteresting, or perhaps
Unimportant?

Any references would be appreciated. [See comment below.]

REPLY [7 votes]: Andrej and Nate have given good introductions to a number of "sophisticated" ideas in computability theory, but I would like to point out something elementary that ought perhaps to be said beforehand.
We have been doing practical computatation for at least seven decades now, so it mystifies me why theoretical compubaility still insists on basing all of its encodings on $\mathbb N$. We can all do actual computation in $\mathbb Q$, but it becomes absurdly infeasible if $p/q$ has to be encoded as $2^p 3^q$. This is unnecessary -- why do it?
The axioms for $\mathbb N$ use zero, successor and recursion. If we simply replace unary successor in this with a binary operation (pairing), written $[-|-]$, then we obtain a type $\mathbb T$ of finite binary trees that is sufficient to encode a large variety of computational objects in a way that is computationally feasible.  From the definition it clearly satisfies
$$  {\mathbb T} \equiv {\mathbb 1} + {\mathbb T} \times {\mathbb T}, $$
whilst also
$$ {\mathbf{List}} ({\mathbb T})  \equiv {\mathbb T}  $$
by, for example,  $[a,b,c,d] \equiv [a|[b|[c|[d|0]]]]$.
This form of encoding has been used in functional and logic programming languages at least since Lisp.   Given that it differs so little from the natural numbers I don't understand why it is not adopted in coomputability theory.   Of course $\mathbb T$ is isomorphic to $\mathbb N$, but the functions are ridiculously infeasible.