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?

There are Turing-complete systems like Jot where every natural number can be mapped to 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?