Model theory

Model theory studies how formal sentences and theories relate to the mathematical structures that make them true. It investigates which properties structures share and what theories can express about them.

First-order logic: A formal logic with quantifiers over individual objects and predicates describing relations among them. Its languages provide the standard syntax whose interpretations model theory studies.
Compactness theorem: A theorem stating that a set of first-order sentences has a model if every finite subset has a model. It turns finite satisfiability into existence of a model for the whole theory.
Definable set: A subset of a structure specified by a formula, possibly using parameters from that structure. Definability measures which subsets a language can describe inside a model.
Second-order logic: A logic that quantifies over relations, properties, or sets as well as individual objects. Its greater expressive power comes with different completeness and compactness behavior.
Alfred Tarski: A Polish-American logician who shaped modern semantics, truth theory, and model theory. His definition of truth for formal languages established a rigorous semantics for structures.
Formal language: A set of finite strings built from an alphabet according to specified formation rules. Logical formulas are strings whose symbols receive interpretations in structures.
Löwenheim–Skolem theorem: A theorem giving models of specified infinite cardinalities for first-order theories with infinite models. It shows first-order theories often admit models of different sizes.
Quantifier elimination: A property of a theory in which every formula is equivalent, modulo the theory, to a quantifier-free formula. It simplifies descriptions of definable sets and can make theories easier to analyze.
Infinitary logic: A family of logics that permits conjunctions, disjunctions, or quantifier strings of infinite length. It relaxes first-order syntax to express properties that ordinary formulas cannot.
Leopold Löwenheim: A German mathematician whose work produced an early theorem on the sizes of models of logical sentences. His result became one half of the Löwenheim–Skolem theorem.

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