I am just curious about something. I think this is trivially true, but I wanted to be sure.
Suppose we have two elementary equivalent $\mathcal{L}$-structures $\mathfrak{A}$ and $\mathfrak{B}$. Also, suppose $\text{Th}(\mathfrak{A})$ has built-in Skolem functions, namely, for all $\mathcal{L}$-formulas $\phi(v,w_1,\cdots,w_n)$ there is a function symbol $f$ such that $\text{Th}(\mathfrak{A})\vDash \forall \bar{w} ((\exists v \phi(v,\bar{w}))\rightarrow \phi(f(\bar{w}),\bar{w}))$. Can we say the same for $\mathfrak{B}$, in this case?
Best Answer
Yes. Specifically, by elementary equivalence the same function symbols serve as Skolem functions in $\mathfrak{B}$: "$\forall\overline{w}((\exists v\phi(v,\overline{w}))\rightarrow\phi(f(\overline{w}),\overline{w}))$" is a first-order sentence, so if it's true in $\mathfrak{A}$ it's true in $\mathfrak{B}$.