In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language

a unique, called its Gödel number. The concept was used by Kurt Gödel for the proof of his incompleteness theorems. (Gödel 1931)natural numberThe natural numbers, N, form a commutative

monoidunder addition (identity element zero), or multiplication (identity element one). A submonoid of N under addition is called a numerical monoid. The positive integers, N ∖ {0}, form a commutative monoid under multiplication (identity element one).The set of integers Z forms a commutative

ringunder addition and multiplication

What happens if I replace a *unique natural number* that form a commutative Monoid with *the set of integers* Z that form a commutative Ring ?

The more specific question should be: What happens **if I use** (not if I replace) a unique natural number that form a commutative Monoid **as** (not *with*) the set of integers Z that form a commutative Ring ?

**Monoid $\Rightarrow$ Ring**

**P.S:** I don't want replace formally but I do not want to really change, but to make it believe that it is as if I had replaced.

withthe commutative semiring structure and gotoMonoidal structure onlyor alsoyou can beginsfrommonoidal structure and gotocommutative ring structure ? Why you should start from a higher structure only and go down to the lower one then ? Can you start from lower and go to higher ? $\endgroup$ – Peter Long Jan 2 at 15:40fromthereto upthe commutative semiring structure again $\endgroup$ – Peter Long Jan 2 at 16:23clearly and preciselystate the issues you see, rather than just throwing out complicated phrases? There may be a good question here (although, more appropriate for MSE) but as currently phrased it is too unclear to admit an answer (given that you don't find Joel's answer satisfactory for some reason). $\endgroup$ – Noah Schweber Jan 2 at 18:07