Ultraproducts and Compactness

Ultraproducts provide a model-theoretic construction that combines a family of structures into a single structure using an ultrafilter. This construction gives an algebraic proof of the compactness theorem and produces nonstandard models.

Ultrafilters

Definition4.8Ultrafilter

An ultrafilter $\mathcal{U}$ on a set $I$ is a maximal filter on the power set of $I$ . Equivalently, $\mathcal{U} \subseteq \mathcal{P}(I)$ satisfies: (1) $\emptyset \notin \mathcal{U}$ , $I \in \mathcal{U}$ ; (2) $A, B \in \mathcal{U} \implies A \cap B \in \mathcal{U}$ ; (3) $A \in \mathcal{U}$ and $A \subseteq B \implies B \in \mathcal{U}$ ; (4) for all $A \subseteq I$ , either $A \in \mathcal{U}$ or $I \setminus A \in \mathcal{U}$ .

Definition4.9Ultraproduct

Given $\mathcal{L}$ -structures $\{\mathcal{M}_i\}_{i \in I}$ and an ultrafilter $\mathcal{U}$ on $I$ , the ultraproduct $\prod_{i \in I} \mathcal{M}_i / \mathcal{U}$ has domain $\prod M_i / \sim$ , where $(a_i) \sim (b_i)$ iff $\{i : a_i = b_i\} \in \mathcal{U}$ . Functions and relations are defined coordinate-wise modulo $\mathcal{U}$ .

Los's Theorem

ExampleNonstandard Integers

Let $\mathcal{U}$ be a non-principal ultrafilter on $\mathbb{N}$ . The ultrapower $\mathbb{N}^* = \mathbb{N}^\mathbb{N}/\mathcal{U}$ is a nonstandard model of Peano arithmetic. It contains "infinite" natural numbers, such as the equivalence class of $(1, 2, 3, \ldots)$ , which is larger than every standard natural number.

RemarkLos's Theorem and Compactness

Los's Theorem states that $\prod \mathcal{M}_i / \mathcal{U} \models \sigma$ iff $\{i : \mathcal{M}_i \models \sigma\} \in \mathcal{U}$ . This gives an algebraic proof of the compactness theorem: if every finite subset of a theory $T$ has a model, choose models $\mathcal{M}_\Delta$ for each finite $\Delta \subseteq T$ , and use an ultrafilter on the directed set of finite subsets. The ultraproduct models all of $T$ .

Applications

Definition4.10Pseudo-Finite Structure

A structure $\mathcal{M}$ is pseudo-finite if every first-order sentence true in $\mathcal{M}$ is true in some finite structure. Equivalently, $\mathcal{M}$ is elementarily equivalent to an ultraproduct of finite structures. Pseudo-finite fields, groups, and graphs play an important role in model theory and additive combinatorics.

ExampleAx's Theorem

Every injective polynomial map $f: \mathbb{C}^n \to \mathbb{C}^n$ is surjective. Proof sketch: For finite fields $\mathbb{F}_q$ , every injective map $\mathbb{F}_q^n \to \mathbb{F}_q^n$ is surjective (pigeonhole). The statement "every injective polynomial of degree $\leq d$ is surjective" is first-order and true in all $\mathbb{F}_q$ . By Los's theorem applied to an ultraproduct $\prod \mathbb{F}_q / \mathcal{U}$ , it holds in a pseudo-finite field of characteristic zero, hence in all algebraically closed fields of characteristic zero by completeness of $\mathrm{ACF}_0$ .