In Godel's first incompleteness theorem, what is the appropriate notion of interpretation function?

Question

Wikipedia states Godel's first incompleteness as follows.

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, [1] but not provable in the theory (Kleene 1967, p. 250).

The notion of 'expressing elementary arithmetic' suggests an interpretation function. In particular, let $\mathsf{LA}$ denote the first-order language of arithmetic and $\mathsf{BA}$ denote the 'basic arithmetical truths' mentioned in the quote. Then Godel's first theorem says that, if $(T,L)$ is a theory/language pair such that there exists an interpretation function $$f : (\mathsf{BA},\mathsf{LA}) \rightarrow (T,L)$$

then $T$ cannot be all three of:

1. effectively generated
2. consistent
3. negation-complete.

What is the appropriate notion of interpretation function here?

Answer 1

The relevant notion of "capable of expressing elementary arithmetic" could be formalized with something like:

1. There's a primitive recursive family of numeral predicates $\psi_0(x), \psi_1(x), \ldots, \psi_n(x), \ldots$ such that $ T\vdash\exists! x.\psi_i(x)$ for each $i$ and $T\vdash \neg\psi_i(x)\lor \neg\psi_j(x)$ for $i\ne j$.

2. For each primitive recursive function $f$ of $k$ variables, there's a formula $\phi(x_1,\ldots,x_k,y)$ such that $$ T\vdash \psi_{n_1}(x_1) \land \cdots \land \psi_{n_k}(x_k) \to \bigl(\phi(x_1,\ldots,x_k,y)\leftrightarrow \psi_{f(n_1,\ldots,n_k)}(y)\bigr) $$ for all $n_1,\ldots,n_k$.

possibly with some additional technical restrictions (e.g., the $\omega$-consistency requirement in Gödel's original work would correspond to some additional conditions on what can be proved about the $\psi_i$s).