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.