38 Logic as a Tool We give the set C(B, F) a precise mathematical meaning by defining it in two different, yet eventually equivalent, ways 1.4.3.2 Top-down closure construction Definition 20 A set C ⊆ U is: closed under the operation f ∈ F, such that f : U n → U , if f (x1 , , xn ) ∈ C for every x1 , , xn ∈ C closed, if it is closed under every operation f ∈ F inductive, if B ⊆ C and C is closed Remark 21 The elements of B can be regarded as constant (0-argument) functions in U , and the condition B ⊆ C can therefore be subsumed by closedness Proposition 22 Intersection of any family of inductive sets is an inductive set Proof I leave this as an exercise Definition 23 C ∗ is the intersection of the family of all inductive sets By Proposition 22, C ∗ is the smallest inductive set 1.4.3.3 Bottom-up inductive construction Definition 24 A construction tree for an element x ∈ U is a finite tree T (x), every node of which is labeled with an element of U and the successors of every node are ordered linearly, satisfying the following conditions: Every leaf in T (x) is labeled by an element of B If a node in T (x) labeled with y has k successors labeled by elements listed in the order of successors y1 , , yk , then there is a k -ary operation f ∈ F such that y = f (y1 , , yk ) The root of T (x) is labeled by x Definition 25 The height of a finite tree is the length (number of nodes minus 1) of the longest path in the tree Definition 26 The rank of an element x ∈ U is the least height r(x) of a construction tree for x if it exists; otherwise, the rank is ∞ Definition 27 We define a hierarchy of sets C0 ⊆ C1 ⊆ · · · ⊆ C∗ as follows Cn is the set of all elements of U with rank ≤ n Cn C∗ := n∈N