Past exam of the model theory course of the University of Oxford / 2011 / 3 / b / iii

The Łoś theorem states that, for every first-order formula $\phi (\mathbf{x})$ and product representatives ${\mathbf{a}}^1,\dots ,{\mathbf{a}}^r$,It extends the defining relation-symbol rule to arbitrary first-order formulas, including quantified ones.

Identify the countable index set with $\mathbb{N}$. A nonprincipal ultrafilter on it contains every cofinite set. Let $p(x)$ be a type over a finite parameter set in the ultraproduct $\mathcal{M}$, finitely satisfiable in $\mathcal{M}$. Because the language and parameter set are countable, enumerate it as ${\phi }_0(x),{\phi }_1(x),\dots$, with chosen product representatives for the parameters. By finite satisfiability and Łoś,Make these large sets decrease while forcing their intersection to be empty:For each index $i$, let $r(i)$ be the largest $n\le i$ with $i\in Y_n$, if one exists. Choose $b_i\in A_i$ satisfying the first $r(i)+1$ first-order formulas; this is possible because $i\in X_{r(i)}$. If no such $n$ exists, choose any $b_i$.

For each fixed $n$, every $i\in Y_n$ has $r(i)\ge n$, so $b_i$ satisfies ${\phi }_n$. Its coordinate truth set contains $Y_n$ and belongs to $\mathcal{U}$. Another use of Łoś gives $\mathcal{M}\models {\phi }_n([\mathbf{b}])$ for every $n$. Thus $[\mathbf{b}]$ realizes the type. This proves the countable saturation of an ultraproduct over omega, and in particular the ultraproduct is omega-saturated. The construction chooses a finite number of requirements at each coordinate; it does not presume the whole infinite type has a witness in any factor.