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
:= a set of function symbols for every i.e., the arity of each function. := a set of relation symbols for every i.e., the arity of each relation. := a set of constant symbols.
Examples of languages:
- Language of groups =
. - Language of rings =
. - Language of pure sets =
. - Language of graphs =
, where is a binary relation symbol.
An
:= a nonempty set called the universe, domain, or underlying set of .- an interpretation of each
i.e., a function for each . - an interpretation for each
i.e, a set for each . - an element
for each .