$\mathcal{L}$ is the language that contains a single binary function symbol $F$. In the earlier parts of this question, we were told to take the $\mathcal{L}$-structure $\mathcal{M}$ with universe $\omega$ (non-negative integers) and the interpretation of $F$ as $F(x,y) = (x+y)^2$.
From Tarski's theorem on strongly undecidable structures, I figured that all I needed to do was to show that the standard model is defined by this model. I was able to show that $\mathcal{M}$ defines the constants 0 and 1, the addition of non-negative integers $(+)$, and the multiplication of non-negative integers $(\cdot)$.
What I don't understand is how the deductive closure of a sentence might change the decidability of the language? My understanding of strongly undefinable structures is that no matter what sentence I add to it, I can always find another that is independent of the axioms. Am I right?
If I'm not, please enlighten me. I only have basic understanding of model theory and predicate logic, and I'm now studying the incompleteness theorems. If there are any good resources where I can read more about this, please share! Thank you!