# Constraints on the order of program semantics given by an enumeration of turing-complete system programs

There are Turing-complete systems like Jot where every natural number represents a valid program. This results in a Gödel numbering.

Now, if the semantics of the programs were, say

| N | Semantics |
-----------------
| 0 | input + 0 |
| 1 | input + 1 |
| 2 | input + 2 |
| 3 | input + 3 |


etc., then the numbering would never reach other kinds of semantics, like multiplication ("input * 0", "input * 1", "input * 2" etc.) and all other possible computable functions.

So, since this seems like a contradiction of the fact that the system is Turing-complete, does that mean that the program number <-> semantics mapping sequence must follow a different pattern that eventually reaches every possible semantics, without having any one pattern going into infinity?

In other words, does a system being Turing-complete set constraints on the (order of) program semantics enumerated by a Gödel numbering? If so, can anything more detailed said about them?

A Tuing-complete programming language has the property that there is a surjection $$c : \mathbb{N} \to \mathsf{Prog}$$ (the set of all valid programs in whatever model of computation you have). We call $$n$$ the Gödel code of the program $$c(n)$$.
Several important theorems rely on having such a map $$c$$, for example the proof that there exists a universal Turng machine. In fact, in those theorems $$c$$ must not only be surjective, but also acceptable. This is a technical condition, and I think it is the one you are asking for (in addition to $$c$$ being surjective). A computable map $$c : \mathbb{N} \to \mathsf{Prog}$$ is acceptable, if it can be used to show that $$\mathsf{Prog}$$ is Turing-complete, i.e., there is a Turing machine $$T$$ (the interpreter) such that $$T(n)$$ "does the same thing" as $$c(n)$$. Inutitively, this says that a Turing machine can actually figure out what $$n$$ encodes. Please look up the exact condition in a book on computability.
There are of course other maps $$\mathbb{N} \to \mathsf{Prog}$$, and if you replace an acceptable surjection $$c$$ with one of them, you will break the proofs of those theorems.