mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, an elementary theory is a

theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...

that involves axioms using only

finitary In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values. In standard mathematics, an operat ...

first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

, without reference to

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

or using any axioms that have

consistency strength In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not p ...

equal to set theory. Saying that a theory is elementary is a weaker condition than saying it is algebraic.

Examples

Examples of elementary theories include: * The theory of

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

** The theory of

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

s ** The theory of

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

s * The theory of

fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...

** The theory of

finite fields In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...

** The theory of

real closed field In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. Def ...

s * Axiomization of Euclidean geometry

Elementary definition In mathematical logic, an elementary definition is a definition that can be made using only finitary first-order logic, and in particular without reference to set theory Set theory is the branch of mathematical logic that studies Set (math ...

* Elementary theory of the reals

References

* Mac Lane and

Moerdijk Moerdijk () is a municipality and a town in the South of the Netherlands, in the province of North Brabant. History The municipality of Moerdijk was founded in 1997 following the merger of the municipalities of Fijnaart en Heijningen, Klunde ...

, ''Sheaves in Geometry and Logic: A First Introduction to Topos Theory,'' page 4. {{mathlogic-stub Mathematical logic

Examples

Related

References