mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

, an elementary definition is a definition that can be made 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 operation ...

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

, and in particular without reference to

set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...

or using extensions such as

plural quantification In mathematics and logic, plural quantification is the theory that an individual variable x may take on ''plural'', as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London ...

. Elementary definitions are of particular interest because they admit a complete proof apparatus while still being expressive enough to support most everyday mathematics (via the addition of elementarily-expressible axioms such as

Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...

(ZFC)). Saying that a definition is elementary is a weaker condition than saying it is algebraic.

Elementary sentence In mathematical logic, an elementary sentence is one that is stated using only finitary first-order logic, without reference to set theory or using any axioms which have consistency strength equal to set theory. Saying that a sentence is elementary ...

Elementary theory Elementary may refer to: Arts, entertainment, and media Music * ''Elementary'' (Cindy Morgan album), 2001 * ''Elementary'' (The End album), 2007 * ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977 Other uses in arts, entertainment, an ...

References

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

Related

References