I came across the following statement:
Given an $\omega$-sequence of $\mathcal{L}$-structures, $(\mathcal{M}_i : i < \omega)$, and a non-principal ultrafilter $\mathcal{U}$ on $\omega$, the ultraproduct of the $\mathcal{L}$-structures,
$$ \mathcal{M} = \prod_{\mathcal{U}} M_i $$
$\aleph_1$-compact, i.e. for any countable (non-empty) collection of non-empty definable sets in $\mathcal{M}$ that has the Finite Intersection Property, the collection has a non-empty intersection.
Perhaps the given statement was incomplete (it was in the middle of a lecture, as an aside), but in its proof, it is assumed that the collection is nested, and the rest of the proof inherently relies on this property.
I wonder if the "nestedness" of the collection is actually a necessary condition. I set out for a proof, but I am unable to arrive at a conclusion. Here is an outline of my attempt:
Let $(F_n)_{n < \omega}$ be the collection given by the assumption, and let $\phi_n$ be the $\mathcal{L}$-formula that defines $F_n$, for each $n < \omega$. Then for any $N < \omega$, by the Finite Intersection Property, we can find a realization $[a_N] \in \bigcap_{n = 0}^{N} F_n$ such that
$$\mathcal{M} \models \bigwedge_{n=0}^{N} \phi_n ([a_N]).$$
By Łoś, we have
$$ \left\{ i \in I : \mathcal{M}_i \models \bigwedge_{n = 0}^{N} \phi_n(a_N(i)) \right\} \in \mathcal{U}. $$
That is where I am stuck at. I reckon that my goal is to perhaps show that
$$\left\{ i \in I : \mathcal{M}_i \models \bigwedge_{n < \omega} \phi_n(a'(i)) \right\} \in \mathcal{U}$$
and complete the proof by Łoś, but I am not entirely certain if that is what I should be pursuing.
Best Answer
It's not a necessary assumption. If $(D_i)_{i\in\omega}$ is a family of sets with the finite intersection property, then let $E_i=\bigcap_{j\le i}D_i$; we have $E_0\supseteq E_1\supseteq ...$ and each $E_i$ is nonempty by the finite intserction property of the $D_i$s. Moreover, if the $D_i$s were definable, so are the $E_i$s, and anything in the intersection of the $E_i$s is in the intersection of the $D_i$s. So we can go from an arbitrary collection to a nested collection, find something in the intersection of the elements of the nested collection, and say that it's also in the intersection of the elements of the original collection.