AXIOMATIC—the calculable subset of the set of propositions of a particular domain of knowledge, such that the set of consequences of this subset (in view of a certain calculable set of calculable operations of inference) is equal to the set of this domain’s propositions.
One of basic features of an axiom in a particular theory is what propositions of the theory can be derived from the axiom. This feature depends upon the other axioms and upon the rules of inference we accept. A full characterization of the axioms can be performed only in the context of all the other axioms and the demonstrative means accepted in the theory. Only the presentation of the axiomatic system defined by the system of axioms, namely the axiomatic, and by the set of operations of inference, makes possible a sufficient grasp of the meaning and role of a particular axiom in the theory.
An axiomatic can be characterized, among other ways, in terms of the features of the axiomatic systems that are its components, are in view of its finitude or independence.
L. Borkowski, Wprowadzenie do logiki i teorii mnogości [Introduction to logic and set theory]; W. Pogorzelski, Elementarny słownik logiki formalnej [An Elementary lexicon of formal logic], Białystok 1992, 33–36.