Definition:Mathematical System

Definition
A mathematical system is a set $$\mathcal S = \left\{{E, O, A}\right\}$$ where:


 * $$E$$ is a non-empty set of elements;


 * $$O$$ is a set of relations and operations on the elements of $$E$$;


 * $$A$$ is a set of axioms concerning the elements of $$E$$ and $$O$$.

Abstract System
A mathematical system $$\mathcal S = \left\{{E, O, A}\right\}$$ is classed as abstract if the elements of $$E$$ and $$O$$ are defined only by their properties as specified in $$A$$.

Concrete System
A mathematical system $$\mathcal S = \left\{{E, O, A}\right\}$$ is classed as concrete if the elements of $$E$$ and $$O$$ are understood as objects independently of their existence in $$\mathcal S$$ itself.

Algebraic System
A mathematical system $$\mathcal S = \left\{{E, O, A}\right\}$$ is classed as algebraic if it has many of the properties of the set of integers.

This is usually because such a system is itself an abstraction of certain properties of the integers.

The distinction between abstract and concrete is of questionable value from a modern standpoint, as it is a moot point, for example, as to whether the set of natural numbers exists independently of Peano's axioms or are specifically defined by them.