= Solution
An <ultraproduct> converts coordinatewise finite information into a single <first-order structure>. Let $(M_i)_{i\in I}$ be nonempty structures in a common <first-order language>, and let $U$ be an <ultrafilter> on $I$. In $\prod_iM_i$, <set>
$$
f\sim_U g\quad\Longleftrightarrow\quad\{i:f(i)=g(i)\}\in U.
$$
The <ultraproduct> $\prod_iM_i/U$ has these <equivalence classes> as its elements. <Functions> are interpreted coordinatewise; a relation holds of classes exactly when it holds on a $U$-large <set> of indices. Intersecting finitely many large agreement <sets> shows that these interpretations are independent of representatives.
The <Łoś theorem>, which explains the construction's usefulness, says
$$
\prod_iM_i/U\models\varphi([f_1],\ldots,[f_r])
\quad\Longleftrightarrow\quad
\{i:M_i\models\varphi(f_1(i),\ldots,f_r(i))\}\in U.
$$
Here is the proof. Induction on terms establishes the coordinatewise interpretation of atomic formulas. Finite intersections handle conjunction, and the <ultrafilter> property handles negation, since precisely one of a <set> and its complement is large. For an <existential quantification>, a witness class gives coordinate witnesses on a large <set>. Conversely, when the coordinate existential truth <set> is large, select a witness in each of those structures and a default element elsewhere. Its class is a witness in the <ultraproduct>. This uses <choice>. Structural induction on formulas now proves the theorem, with universal quantification obtained by negation.
\b[Compactness.] If every finite <subset> of a first-order theory $T$ has a model, take $I$ to be the finite <subsets> $\Gamma\subseteq T$ and choose $M_\Gamma\models\Gamma$. The cones $\{\Gamma:\Delta\subseteq\Gamma\}$ for finite $\Delta$ have the finite intersection property. Extend their filter to an <ultrafilter> by the <ultrafilter lemma>. For each $\varphi\in T$, its cone is large, so the Łoś theorem makes the <ultraproduct> satisfy $\varphi$. This proves the <compactness theorem>, including theories of arbitrary <set> size.
\b[Elementary extensions and nonstandard elements.] An <ultrapower> uses a fixed structure at every coordinate. The diagonal map sending $a$ to the class of the constant <function> $a$ is an <elementary embedding> by the Łoś theorem. For a <nonprincipal ultrafilter> on $\omega$, the class of $i\mapsto i$ in an <ultrapower> of the natural numbers exceeds every standard integer, because each corresponding tail is large. In an <ultrapower> of the real ordered field, the reciprocal of this element is positive and smaller than every positive standard reciprocal integer: an infinitesimal. These examples implement ordinary finite statements in a larger structure without claiming that every external <subset> or second-order assertion transfers.
\b[Saturation.] Such an <ultraproduct> over $\omega$ is countably saturated. To see the actual diagonal argument, list a finitely satisfiable type $\varphi_0(x),\varphi_1(x),\ldots$ with parameters in the <ultraproduct> and fix representatives for those parameters. For each $n$, the <set> $E_n$ of coordinates admitting a simultaneous witness to its first $n$ formulas belongs to $U$ by the Łoś theorem. Arrange that the $E_n$ decrease and put $D_n=E_n\cap\{i:i\geq n\}$. The $D_n$ decrease, belong to $U$, and have empty intersection. At coordinate $i$, choose a witness to the first $m(i)$ formulas, where $m(i)$ is the largest $n$ with $i\in D_n$, or choose a default element if there is no such $n$. For each fixed $n$, $m(i)\geq n$ on the large <set> $D_n$. Therefore the resulting class realizes every formula. This proves <countable saturation of an ultraproduct over omega>; it does not assert saturation at all larger cardinalities.
\b[Algebraic limits.] Take finite fields $\mathbb F_p$ indexed by the primes and a nonprincipal <ultrafilter>. Their <ultraproduct> is a field of characteristic zero: for every fixed positive integer $n$, only finitely many primes divide $n$, so $n\cdot1\ne0$ on a large <set>. Every first-order sentence true in all finite fields holds in this infinite field, and more generally so does any sentence true for all but finitely many of the factors. This is a way to create an infinite field retaining uniform first-order information from finite fields. It does not make the field algebraically <closed>: the first-order sentence that every element is a square already fails in infinitely many of the factors.
\b[Large <cardinals>.] A <countably complete ultrafilter> gives a well-founded membership <ultrapower> of the universe. If it had an infinite descending chain represented by $f_n$, each <set>
$$
A_n=\{i:f_{n+1}(i)\in f_n(i)\}
$$
would be large. <Countable> completeness makes $\bigcap_nA_n$ nonempty, giving an actual infinite membership descent at any index in the intersection, contrary to <foundation>. The <Mostowski collapse> therefore gives a transitive target and an <ultrapower embedding> $j:V\to M$.
For a measure on a <measurable cardinal> $\kappa$, every <function> $\kappa\to\alpha$ with $\alpha<\kappa$ is constant on a large <set>: otherwise intersect the fewer than $\kappa$ complements of all its fibres. Consequently $j$ fixes the <ordinals> below $\kappa$. The class of the identity <function> is below $j(\kappa)$ and above every constant class below $\kappa$, since all tails are large. Thus $j(\kappa)>\kappa$ and its <critical point> is $\kappa$. <Fine ultrafilters> on $P_\kappa(\lambda)$ that are <normal ultrafilters on small subsets> and <kappa-complete> yield the stronger embeddings witnessing <supercompact cardinals>.
These applications share a single principle: \b[an <ultraproduct> preserves precisely the first-order information seen by its <ultrafilter>]. Principal <ultrafilters> merely select a factor; nonprincipal ones can produce new sizes, elements and saturation. Existence of the needed <ultrafilters> and selection of coordinate witnesses are set-theoretic assumptions, and changing the <ultrafilter> may change the resulting model.
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