Axiomatics through the Metric Space Axioms

In this article we discuss some elementary concepts in axiomatics, which is the study of axioms. The axiomatic system we use is that of a metric space, which will be familiar to many readers. Metric spaces were introduced by the French mathematician Maurice Fréchet in 1906 as a generalization of the real numbers and the “distance function” f (x , y) = |x − y|. The aim is to construct limits, continuity, openness and eventually differential calculus on this generalized base alone; the results will then apply to a wide class of spaces such as R2 without little particular work. Here we are only interested in metric spaces as an example of an axiomatic system; that is, a system which behaviors according to clear, welldefined rules. What do we desire in an axiom set? Intuitively, we might want our axiom sets to be as small as possible. In particular, if one axiom in an axiom set can be logically deduced from the others we might be inclined to remove it, as it serves no purpose and just makes our set less “pure” than it could be. If a specific axiom can’t be deduced from the others then we say that that axiom is independent. If every axiom in a set is independent, we say the whole set is independent. Another, more important, requirement of our set of axioms is that it does imply contradictory statement. Such an axiom set would be effectively useless. An axiom set that isn’t inherently contradictory is called consistent. The traditional definition of a metric space provides suitable grounds to discuss these and other concepts, and to illustrate techniques for proving properties of axiom sets.