Model theory
Home

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 L is given by specifying the following data:

  • F := a set of function symbols
  • nf:N for every fF i.e., the arity of each function.
  • R := a set of relation symbols
  • nr:N for every rR i.e., the arity of each relation.
  • C := a set of constant symbols.

Examples of languages:

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

An L-structure M is given by the following data:

  1. M := a nonempty set called the universe, domain, or underlying set of M.
  2. an interpretation of each fF i.e., a function fM:MnfM for each f.
  3. an interpretation for each rR i.e, a set RMMnr for each r.
  4. an element cMM for each cC.

Author: Vaibhav Karve

Created: 2024-02-04 Sun 21:01

Validate