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

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...

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...

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

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)...

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...

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...

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 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...

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...

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...

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...

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

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...

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 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...

Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated...

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...

A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences. The semantic conception...

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...

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

formalism". Furthermore in the theory, it is almost immediately observable that interpretations (in the sense of model theory) are presented in terms of truth-values...

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

theorem. These classes can serve as inner models for various axiomatic systems such as ZFC or Morse–Kelley set theory. Universes are of critical importance...

science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives...