Axioms are the foundation of a theory, that from which all its claims are derived. What then grounds or justifies the axioms themselves? In practice, axioms are justified in large part by their implications. This may sound circular, but isn't on reflection. As we start theorizing in some particular domain of inquiry, we already have firm ideas about some truths and falsities, and we want to formulate our axioms so that they confirm these antecedent commitments. This approach is captured by the term axiomatization. We are to axiomatize our antecedent commitments, that is, we are to formulate a small set of axioms from which we can elegantly derive the much larger and messier set of propositions we hold true antecedently (including negations of propositions we hold false antecedently). Of course, such an axiomatization is successful only if the set of chosen axioms does not permit derivation of a contradiction.

Somewhat paradoxically, axioms are then justified not by appeal to something further "upstream," but by their implications. Relatedly, axioms may not be especially evident or intuitive. In fact, an axiom may be quite unituitive. What matters is that is "works": that it, together with the other axioms chosen, allows us to derive, without contradiction, what we take to be true.