This is a follow-up to my previous model theory question, here: Is addition definable from successor and multiplication?. I asked whether addition is definable by a first-order formula in the structure $(\mathbb{N};\times,S,0,1)$ in that question, and found that that question was asked previously. However, now I have a different question. Is addition a term function in that structure? A term function is a function composed from projection functions and the functions and constants that are in the structure. I think it isn't, but I want a proof that it is not.
2026-03-29 11:20:08.1774783208
Is addition a term-function in this structure?
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No, there is not.
Suppose $t$ is a term such that $t(x,y)\equiv x+y$. WLOG we may assume that $t$ has no instance of "$0\times$" or "$\times 0$" occurring in it. Clearly both variables $x$ and $y$ must actually occur in $t$. But now we can show by induction on complexity that $\forall x,y[t(x,y)\ge xy]$, which isn't true for $x+y$.