Axiom schema

In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. For this reason, an axiom schema is used. Formally, an axiom schema is a set (usually infinite) of well formed formulae, each of which is taken to be an axiom. Often, this set is constructed recursively. A well known axiom schema is the axiom schema of replacement.

There is debate among metamathematicians as to whether an axiomatic system containing an axiom schema should be considered elegant. Some logicians thus prefer, if possible, to use a finite number of axioms.