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...

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...

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...

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)...

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

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...

is structurally complete if every admissible rule is derivable. A theory is model complete if and only if every embedding of its models is an elementary...

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...

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

Glossary of set theory Class (set theory) List of set theory topics Relational model – borrows from set theory Venn diagram Elementary Theory of the Category...

The more modern field of model theory refers to mathematical structures. The relationship between an axiom systems and the models that correspond to it is...

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...

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

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...

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...

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 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...

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...

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...

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 c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the...

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...

In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing...

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...

In computability theory, a system of data-manipulation rules (such as a model of computation, a computer's instruction set, a programming language, or...

{\displaystyle n} , and output a prime larger than n {\displaystyle n} . In category theory, truth values appear as the elements of the subobject classifier. In particular...