In
mathematical logic
Mathematical logic is the study of 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 formal ...
, an elementary definition is a definition that can be made using only
finitary 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 quanti ...
, 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 concer ...
or using extensions such as
plural quantification. 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 (ZFC)).
Saying that a definition is elementary is a weaker condition than saying it is
algebraic.
Related
*
Elementary sentence
*
Elementary theory
References
* Mac Lane and Moerdijk, ''Sheaves in Geometry and Logic: A First Introduction to Topos Theory,'' page 4.
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Mathematical logic