# Model theory

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## Table of Contents

- Book
- David Marker's Model Theory: an introduction.
- Book
- Open logic project link to built pdf (chapters 14 and 15)
- Wikipedia
- First order logic link

## 1 Basic concepts

Syntax and semantics of expressions → first-order logic (FOL) → theory and metatheory of FOL.

Expressions of FOL := terms and formulas.

Terms := variables | constant-symbols | function-symbols.

Formulas := predicate-symbols and terms (combined recursively using logical connectives and quantifiers).

Semantics deals with the concept of satisfaction in a structure. Using satisfiability, it then defines other notions like validity and entailment.

## 2 First-order languages

## 3 Structures and theories

A *structure* is a set that we wish to study equipped with collection
of distinguished functions, relations, and elements. We also choose a
*language* (which is a set of symbols) to express statements about the
structure.

A *language* \(\mathcal{L}\) is given by specifying the following data:

- \(F\) := a set of function symbols
- \(n_f : \mathbb{N}\) for every \(f \in F\) i.e., the
*arity*of each function. - \(R\) := a set of relation symbols
- \(n_r : \mathbb{N}\) for every \(r \in R\) i.e., the
*arity*of each relation. - \(C\) := a set of constant symbols.

Examples of languages:

- Language of groups = \(\{\cdot, e\}\).
- Language of rings = \(\{+, -, \cdot, 0, 1\}\).
- Language of pure sets = \(\emptyset\).
- Language of graphs = \(\{R\}\), where \(R\) is a binary relation symbol.

An \(\mathcal{L}\)-*structure* \(\mathcal{M}\) is given by the following data:

- \(M\) := a nonempty set called the universe, domain, or underlying set of \(\mathcal{M}\).
- an interpretation of each \(f\in F\) i.e., a function \(f^\mathcal{M} : M^{n_f} \rightarrow M\) for each \(f\).
- an interpretation for each \(r \in R\) i.e, a set \(R^\mathcal{M}\subseteq M^{n_r}\) for each \(r\).
- an element \(c^\mathcal{M} \in M\) for each \(c\in C\).