**Łoś's theorem:**

*If $F$ is an ultrafilter on $I$, $M_i$ is a model with domain $A_i$, then for any formula $\phi$ of L and any sequence $f/F \in (\prod A_i / F)^\omega$*
$$\prod M_i/F \vDash_{f/F} \phi \qquad \text{iff} \qquad \{i \in I: M_i \vDash_{f(i)} \phi \} \in F$$

Bell and Slomson's book *Models and Ultraproducts: An Introduction* suggests to interpret it as: a formula holds in $\prod M_i / F$ iff it holds in 'almost all' it's factors. However it seems clear that this can't be right in some cases. For take $I$ to be the set of natural numbers and let $F$ be the ultrafilter generated by $1$.Then, let the sequence $f$ be defined as: $(f_1, f_2, f_3,...)$ where each $f_i=(1, i+1, i+2, i+3, ...)$. And finally $\phi(v_0, v_1)$ is $v_0=v_1$. In this case $\{i \in I: M_i \vDash_{f(i)} \phi \}$ is just the set $\{1\}$, and so the intuition doesn't hold.

Sure, in general if the set $I$ is of cardinal $k$ the proposed intuition will hold in $2^k$ cases and won't in only $k$ cases; but I was wondering if anyone had a better intuition of how to understand this result.