 Condition (β) is crucial since it ensures minimality for the set of natural numbers, which accounts for the validity of proofs by mathematical induction. \(N = \phi_{o}\{1\}\) is read: \(N\) is the chain of singleton {1} under the function φ, that is, the minimal closure of {1} under the function φ. In general, one considers the chain of a set \(A\) under an arbitrary mapping γ, denoted by \(\gamma_{o}(A)\); in his booklet Dedekind developed an interesting theory of such chains, which allowed him to prove the Cantor-Bernstein theorem. The theory was later generalized by Zermelo and applied by Skolem, Kuratowski, etc.