Elementary Equivalence and Types

Two of the most important concepts in model theory are elementary equivalence (when two structures satisfy the same sentences) and types (consistent collections of formulas that describe potential elements).

Elementary Equivalence

Definition4.4Elementary Equivalence

Two $\mathcal{L}$ -structures $\mathcal{M}$ and $\mathcal{N}$ are elementarily equivalent, written $\mathcal{M} \equiv \mathcal{N}$ , if for every $\mathcal{L}$ -sentence $\sigma$ : $\mathcal{M} \models \sigma \iff \mathcal{N} \models \sigma$ .

Definition4.5Elementary Substructure

$\mathcal{M}$ is an elementary substructure of $\mathcal{N}$ , written $\mathcal{M} \preceq \mathcal{N}$ , if $\mathcal{M} \subseteq \mathcal{N}$ and for every formula $\varphi(x_1, \ldots, x_n)$ and $a_1, \ldots, a_n \in M$ : $\mathcal{M} \models \varphi(a_1, \ldots, a_n) \iff \mathcal{N} \models \varphi(a_1, \ldots, a_n)$ .

Example$(\mathbb{Q}, <) \equiv (\mathbb{R}, <)$?

No! $(\mathbb{R}, <)$ satisfies the completeness axiom (every bounded set has a supremum), but this is not first-order expressible. However, both $(\mathbb{Q}, <)$ and $(\mathbb{R}, <)$ are elementarily equivalent to each other as dense linear orders without endpoints. By Cantor's theorem, any two countable dense linear orders without endpoints are isomorphic, so all countable models of this theory are isomorphic.

Types

Definition4.6Complete Type

Let $T$ be a complete $\mathcal{L}$ -theory and $A \subseteq M$ for a model $\mathcal{M} \models T$ . A complete $n$ -type over $A$ is a maximal consistent set $p(x_1, \ldots, x_n)$ of $\mathcal{L}(A)$ -formulas. The set of all complete $n$ -types over $A$ is denoted $S_n^T(A)$ or simply $S_n(A)$ .

Definition4.7Type Space Topology

$S_n(A)$ carries the Stone topology, where basic open sets are $[\varphi] = \{p \in S_n(A) : \varphi \in p\}$ . This makes $S_n(A)$ a compact, Hausdorff, totally disconnected space (a Stone space). The topology of the type space encodes deep structural information about the theory.

RemarkIsolation and Omitting Types

A type $p \in S_n(A)$ is isolated if it is an isolated point of $S_n(A)$ , equivalently, some single formula $\varphi \in p$ implies all others. The omitting types theorem states that a countable theory can omit a non-isolated type in some countable model. This provides a powerful method for constructing models with specific properties.