Skolem function: $f(a_1, \cdots, a_n) \in \bigcap_i B_i$?

Question

Let $\mathcal{A}$ be an $\mathcal{L}$-structure and let $\{\mathcal{B_i}\}_{i \in I}$ be a collection of elementary substructures.

Assume $\mathcal{A}$ has definable Skolem functions.

Given a formula $\phi(v_0,v_1, \cdots, v_n)$ and given $a_1, \cdots, a_n \in \bigcap_i B_i$, can we guarantee $f(a_1, \cdots, a_n) \in \bigcap_i B_i$ for at least some Skolem function $f$ for $\phi$?

Answer 1

Of course! This is obvious from the fact that elementary substructures are closed under definable functions.

In more detail: If $B_i$ is an elementary substructure of $A$, and $\psi$ defines a function $f^A\colon A^n\to A$, then $\psi$ also defines a function $f^{B_i}\colon B_i^n\to B$, and for all $a_1,\dots,a_n\in B_i$, we have $f^A(a_1,\dots,a_n) = f^{B_i}(a_1,\dots,a_n)\in B_i$. This is true for all $i\in I$, so if $a_1,\dots,a_n\in \bigcap_{i\in I} B_i$, we have $f^A(a_1,\dots,a_n)\in \bigcap_{i\in I} B_i$.

Answer 2

Yes.

For every Skolem function $\exists^{=1} y f(a_1,\dots,a_n)=y$ holds in every elementary substructure, by elementarity the witness has to be the same.