We shall discuss the organization of an axiomatic system first through an example, namely Hilbert's famous axiomatization of elementary geometry. Hilbert tried to organize the axioms into groups that stem from the division of the basic concepts of geometry such as incidence, order, etc. Next, detailed axiomatizations of plane projective geometry and lattice theory are presented, based on the use of geometric constructions and lattice operations, respectively. An alternative organization of an axiomatic system uses existential axioms in place of such constructions and operations. It is discussed, again through the examples of projective geometry and lattice theory, in Section 3.2.
Organization of an axiomatization
(a) Background to axiomatization. To define an axiomatic system, a language and a system of proof is needed. The language will direct somewhat the construction of an axiomatic system that is added onto the rules of proof: there will be, typically, a domain of individuals, i.e., the objects the axioms talk about, and some basic relations between these objects.
When an axiomatic system is developed in every detail, it becomes a formal system. Expressions in the language of the systems are defined inductively, and so are formal proofs. The latter form a sequence that can be produced algorithmically, one after the other.
The idea of a formal axiomatic system is recent, only a hundred years old. Axiomatic systems appeared for the first time in Greek geometry, as known from Euclid's famous book.