A mathematical construction that consists of a collection of things and their definitions, and a finite set of axioms which are to be satisfied by those things. From there, the game is to make assertions based on these definitions and axioms that imply other interesting facts about the system, in other words, the formulation and proof of theorems.

Modern formal mathematics is really the study of axiomatic systems. The first such system, and the only one most people who have had at least a high school education are exposed to, is geometry, first created by Euclid 2000 years ago. There are many others in more advanced branches of mathematics: group theory, topology, number theory and analysis are all at their core axiomatic systems.

Unfortunately, it has been shown that there are severe limitations to this approach. Gödel's theorem has shown that any axiomatic system of scope sufficient to allow the system to make statements about itself (i.e. make metamathematical statements) is incomplete.