Recursive Definitions

The articles [6], [5], [8], [7], [1], [9], [3], [2], and [4] provide the notation and terminology for this paper. We use the following convention: n, k denote natural numbers, x, y, z, y1, y2 denote sets, and p denotes a finite sequence. Let D be a set, letp be a partial function fromD to N, and letn be an element of D. Thenp(n) is a natural number. In this article we present several logical schemes. The scheme RecExdeals with a set A and a ternary predicateP , and states that: There exists a functionf such that domf = N and f (0) = A and for every element n of N holdsP [n, f (n), f (n+1)] provided the parameters meet the following requirements: • For every natural number n and for every set x there exists a set y such thatP [n,x,y], and • For every natural number n and for all setsx, y1, y2 such thatP [n,x,y1] andP [n,x,y2] holdsy1 = y2. The schemeRecExDdeals with a non empty set A , an element B of A , and a ternary predicate P , and states that: There exists a functionf from N into A such thatf (0) = B and for every element n of N holdsP [n, f (n), f (n+1)] provided the parameters have the following property: • For every natural number n and for every element x of A there exists an element y of A such thatP [n,x,y]. The schemeLambdaRecExdeals with a set A and a binary functor F yielding a set, and states that: There exists a functionf such that domf = N and f (0) = A and for every element n of N holds f (n+1) = F (n, f (n)) for all values of the parameters. The schemeLambdaRecExDdeals with a non empty set A , an elementB of A , and a binary functorF yielding an element of A , and states that: There exists a functionf from N into A such thatf (0) = B and for every element n of N holds f (n+1) = F (n, f (n)) for all values of the parameters. The schemeFinRecExdeals with a set A , a natural number B, and a ternary predicate P , and states that: