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...

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

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

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)...

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements that are not...

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...

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

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

In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the...

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

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

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

levels of his model of hydrogen rather than the orbital frequency. Bohr completed his PhD in 1911 with a thesis 'Studies on the Electron Theory of Metals'...

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...

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...

In model theory, a subfield of mathematical logic, an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula...