Model Theory

Model theory is a branch of mathematics and logic that studies relationships between formal languages and the structures in which their statements are interpreted. Its central questions concern which structures satisfy a given collection of axioms, which properties can be expressed in a language, and how structures can be classified through those properties. Most classical model theory uses first-order logic, although other logical frameworks are also studied. A model here is a structure satisfying specified statements, not necessarily an approximation of a physical system. (plato.stanford.edu)

Languages, structures, and truth

A formal language specifies logical symbols together with a signature of constant, function, and relation symbols. A structure supplies a nonempty domain and interprets those symbols as elements, operations, and relations on that domain. For example, a language containing addition, multiplication, zero, and one can describe fields, while an order relation can describe ordered sets. First-order quantifiers range over domain elements, rather than arbitrary subsets of the domain. (math.berkeley.edu)

The distinction between syntax and semantics is fundamental: formulas are symbolic expressions, whereas their truth depends on interpretation. Tarski-style semantics defines satisfaction recursively, beginning with atomic formulas and proceeding through logical connectives and quantifiers. The notation

means that the formula $\varphi$ holds in the structure $M$ when its free variables receive the tuple $\bar a$. A sentence has no free variables. A theory $T$ is a collection of sentences, and $M\models T$ means that every sentence in $T$ is true in $M$. (plato.stanford.edu)

Comparing models

Two structures are elementarily equivalent if they satisfy exactly the same first-order sentences. This is weaker than isomorphism, which requires a bijection preserving all the named structure. An elementary embedding preserves the truth of every formula, including formulas with parameters; an elementary substructure is a substructure whose inclusion map has this property. (math.berkeley.edu)

A consistent theory is complete if, for every sentence in its language, it entails that sentence or its negation. Completeness therefore concerns agreement among models about sentences, not uniqueness of a model. A theory is categorical in a cardinality if it has exactly one model of that size up to isomorphism. These distinctions separate logical information from structural identity and underpin the classification of models. (plato.stanford.edu)

Fundamental theorems

Gödel’s completeness theorem connects semantic consequence with formal proof: a sentence follows from a first-order theory precisely when it is derivable in a sound and complete proof calculus. This is distinct from completeness of a particular theory. It does not conflict with Gödel’s incompleteness theorems, which concern limitations of effectively axiomatized theories capable of expressing sufficient arithmetic. (math.berkeley.edu)

The compactness theorem states that a first-order theory has a model if every finite subset has a model. For example, add a constant $c$ to the complete theory of the natural numbers and require $c>n$ for every standard numeral $n$. Every finite collection of these requirements is satisfiable. Compactness consequently gives a model containing an element larger than every standard numeral, illustrating how nonstandard models arise. (math.berkeley.edu)

The Löwenheim–Skolem theorems constrain what first-order theories can determine about size. In a countable language, every theory with an infinite model has a countably infinite model. Upward versions provide models in arbitrarily large infinite cardinalities, subject to the language-size bound. Thus a first-order theory with infinite models cannot characterize a single infinite structure up to isomorphism across all cardinalities. (math.berkeley.edu)

Definability and quantifier elimination

A definable set consists of the tuples satisfying a formula in a structure, possibly using parameters. Model theory investigates these sets as mathematical objects in their own right. A theory has quantifier elimination when every formula is equivalent, modulo the theory, to a quantifier-free formula in the same language. This property reduces questions involving quantified statements to the basic relations and operations. (math.berkeley.edu)

Algebraically closed fields admit quantifier elimination in the language of rings. Real closed fields admit it in the language of ordered rings, linking definability to conditions involving polynomial equations and inequalities. In particular, real closed fields are o-minimal: every definable subset of the underlying line is a finite union of points and intervals. This gives a precise restriction on the complexity of one-dimensional definable sets. (math.berkeley.edu)

Types, classification, and development

A type describes a possible tuple through a consistent collection of formulas. A complete type specifies every formula or its negation over a chosen parameter set. A saturated model realizes all types over parameter sets smaller than a specified cardinal bound. Types and saturation support methods for comparing models and analyzing their internal structure. (math.berkeley.edu)

Classification theory studies how logical restrictions organize families of models. Stability, for example, can be formulated through bounds on the number of types over parameter sets. The subject developed from foundational logic into a discipline with substantial connections to algebra and algebraic geometry. During the 1940s, Alfred Tarski, Anatoly Maltsev, and Abraham Robinson helped establish the use of logical methods in mathematics; Tarski proposed the name “theory of models” in 1954. (plato.stanford.edu)