A modern example: Noam Chomsky's axiomatization of Linguistics [to recall another ongoing discussion in this list: Where are you, Mr McCarthy, to "see" him?]
<q> The basic system of description that we shall consider is a system G of the following form: G is a semi-group under concatenation [see below] with strings in a finite set V of symbols as its elements, and I as the identity element. V is called the "vocabulary" of G. V = V_T U V_N (V_T, V_N disjoint), where V_T is the "terminal vocabulary" and V_N the "nonterminal vocabulary." A two-place relation -> is defined on elements of G, read "can be rewritten as." This relation satisfies the following conditions:
AXIOM 1. -> is irreflexive.
[AXIOMs 2-4 omitted, due to symbols not available in the ASCII table]
</q> Information and Control, 2,1959, p. 141.
I recall that I have read in a Greek book on the Middle Ages Philosophy that some Catholic Theologians had tried to axiomatize Theology. Details when I locate the book.