Model Theory

Abstract

Model theory studies the relationship between formal languages (syntax) and their interpretations in mathematical structure (semantics). It focuses on the classification of structures satisfying given axioms.

Definition

Model theory investigates mathematical structures $\mathcal{M}$ via sentences $\varphi$ in a formal language $\mathcal{L}$. The central relation is logical satisfaction, denoted $\mathcal{M}\models \varphi$, defined via Tarski Semantics.

Key Concepts

• Model Structure: A non-empty set equipped with functions, relations, and constants corresponding to the signature of $\mathcal{L}$.
• theory: A set of sentences $T$. A structure $\mathcal{M}$ is a model of $T$ if $\mathcal{M}\models \psi$ for all $\psi \in T$.
• Elementary Equivalence: Two structures $\mathcal{M},\mathcal{N}$ are elementary equivalent ($\mathcal{M}\equiv \mathcal{N}$) if they satisfy the same first-order sentences.

Fundamental Theorems (First-Order)

1. Compactness Theorem: A set of sentences is satisfiable iff every finite subset is satisfiable.
2. Löwenheim-Skolem Theorems: If a countable theory has an infinite model, it has models of every infinite cardinality (both downward and upward).
3. Completeness Theorem: Syntactic derivability coincides with semantic consequence ($\Gamma ⊢\varphi ⟺\Gamma \models \varphi$).

Remarks

In categorical logic, a theory corresponds to a structured category (the syntactic category), and a model is a structure-preserving functor to a semantic category (e.g., Set or a specific topos). This generalizes to Homotopy Type Theory, where theories are interpreted in $\infty$-topoi, and the notion of structure respects equivalence rather than strict equality.