Elementary theory
Encyclopedia
In mathematical logic
, an elementary theory is one that involves axioms 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 theory is elementary is a weaker condition than saying it is algebraic
.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, an elementary theory is one that involves axioms using only finitary
Finitary
In mathematics or logic, a finitary operation is one, like those of arithmetic, that takes a finite number of input values to produce an output. An operation such as taking an integral of a function, in calculus, is defined in such a way as to depend on all the values of the function , and is so...
first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
, without reference to set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
or using any axioms which have consistency strength equal to set theory.
Saying that a theory is elementary is a weaker condition than saying it is algebraic
Algebraic theory
In mathematical logic, an algebraic theory is one that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences.Saying that...
.