In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order...

In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing...

Gödel's completeness theorem is about this latter kind of completeness. Complete theories are closed under a number of conditions internally modelling the...

The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is...

Finite model theory is a subarea of model theory. Model theory is the branch of logic which deals with the relation between a formal language (syntax)...

from the theory. A satisfiable theory is a theory that has a model. This means there is a structure M that satisfies every sentence in the theory. Any satisfiable...

in particular domains or under certain conditions. For some theories, a more complete model is known, but for practical use, the coarser approximation...

logic, and particularly in its subfield model theory, a saturated model M is one that realizes as many complete types as may be "reasonably expected" given...

cardinalities. More precisely, for any complete theory T in a language we write I(T, κ) for the number of models of T (up to isomorphism) of cardinality...

Mathematics portal Glossary of set theory Class (set theory) List of set theory topics Relational model – borrows from set theory Venn diagram In his 1925 paper...

mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely characterizing...

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on...

abstract model theory is a generalization of model theory that studies the general properties of extensions of first-order logic and their models. Abstract...

In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements...

Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural...

first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their...

any model M {\displaystyle M} to which it is elementarily equivalent (that is, into any model M {\displaystyle M} satisfying the same complete theory as...

such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements of sets that...

special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that...

standard semantics does not admit an effective, sound, and complete proof calculus. The model-theoretic properties of HOL with standard semantics are also...

property down to equality. A theory T is an o-minimal theory if every model of T is o-minimal. It is known that the complete theory T of an o-minimal structure...

in the proof that there is no free complete lattice on three or more generators. The paradoxes of naive set theory can be explained in terms of the inconsistent...

mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in...

theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what consistent meant...

mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical...

In set theory, the complement of a set A, often denoted by A ∁ {\displaystyle A^{\complement }} (or A′), is the set of elements not in A. When all elements...

complete. In superintuitionistic and modal logics, a logic is structurally complete if every admissible rule is derivable. A theory is model complete...

Courcelle's theorem, an algorithmic meta-theorem in graph theory. The MSO theory of the complete infinite binary tree (S2S) is decidable. By contrast, full...

of ZF hold true in any model of that theory, regardless of the truth or falsity of the axiom of choice in that particular model. The implications of choice...

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through...