# Model theory Home

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

## 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.

## 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:

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

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

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

Created: 2021-04-08 Thu 10:50

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