I'm currently following a textbook on an introduction to first order logic; it's called The Logic Book (6th edition). I couldn't quite understand the clear distinction between an axiom and an axiom schema given by the definition provided in the textbook:
Notice that it says axiom schema and not scheme for I have been informed that there was a clear difference between the two words. If anyone could be of help and educate me about the distinction between an axiom and an axiom schema, that would be of much help. Thank you in advance.
This is an axiom schema: It is the second ZF axiom of set theory.
Called axiom schem of replacement.
$(\forall x)(\exists y)[x\in y \iff (\exists z)((x\in z) \land \phi(x))]$
what is special about it is that an infinite number of formulae can replace $\phi(x)$.
An axiom on the other hand has one formula only:
axiom of extensionality for example:
$(\forall x)(\forall A)(\forall B)(A=B\iff (x \in A \iff x\in B))$