Independence is not a necessary requirement for a system, yet consistency is necessary. An axiomatic system will be called *complete* if no additional axiom can be added to the system without making the new system either *dependent* or *inconsistent*.

A *mathematical model* for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. The existence of a *concrete model** proves the *consistency* of a system.

Models can also be used to show the *independence* of an axiom in the system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is *independent* if its correctness does not necessarily follow from the subsystem.

Two models are said to be isomorphic if one-to-one correspondence can be found between their elements, in a manner that preserves their relationship. An axiomatic system for which every model is isomorphic to another is called *categorial*, and the property of *categoriallity* ensures the *completeness* of a system.

* A model is called *concrete* if the meanings assigned are objects and relations from the real world, as opposed to an *abstract model* which is based on other axiomatic systems.

The first axiomatic system was Euclidean geometry.