Abstract

We first start by clarifying what axiomatizing everything can mean. We then study a famous case of axiomatization, the axiomatization of natural numbers, where two different aspects of axiomatization show up, the model-theoretical one and the proof-theoretical one. After that we discuss a case of axiomatization in a sense opposed to the one of arithmetic, the axiomatization of the notion of order, where the idea is not to catch a specific structure, but a notion. A third mathematical case is then examined, the one of identity, a simple and obvious notion, but that cannot be axiomatized in first-order logic. We then move on to more general notions: the axiomatization of causality and the universe. To end with we deal with an even more tricky question: the axiomatization of reasoning itself. In conclusion we discuss in the light of our investigations the relation between axiomatization and understanding.