 Meanwhile, other authors were exploring the possibilities opened by set theory for the foundations of mathematics. Most important was Dedekind’s contribution (1888) with a deep presentation of the theory of the natural numbers. He formulated some basic principles of set (and mapping) theory; gave axioms for the natural number system; proved that mathematical induction is conclusive and recursive definitions are flawless; developed the basic theory of arithmetic; introduced the finite cardinals; and proved that his axiom system is categorical. His system had four axioms. Given a function φ defined on \(S\), a set \(N \subseteq S\), and a distinguished element \(1 \in N\), they are as follows: