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Successor FunctionA successor function is the label in the literature for what is actually an operation. A function is a many-one or one-one relation allowing the "input" to be of different type than the "output". Example: "the catalog function", with input a catalog number, output a thing on a shelf. The successor operation has a number input and number output (same type). This is critical, since its operational form allows it to define the natural numbers: The natural numbers are the outputs generated by the successor operation. Furthermore, the successor operation generates the primary operations of natural number arithmetic, addition, multiplication, exponentiation, etc. Definition: S(n) = n + 1, for initial input 0. Thus, S(0) = 0 + 1 = 1; S(S(0)) = S( 1) = 1 + 1 = 2; S(S(S(0))) = S(S(1)) = S(2) = 2 + 1 = 3; ...; S(S(S(S(S(S(S(0))))))) = ... = 7; etc. Addition: S(a) = a + 1, a + S(b) = S(a + b). (Note that the new operation on right uses the old one on right.) Multiplication: a x 1 = a, a x S(b) = a x b + a. Exponentiation: b0 = 1, bS(e) = (bez) x b. The operational character of "successor" evokes the Closure property. The natural numbers are closed under the successor operation; addition of natural numbers is closed under the successor operation; similarly, for multiplication and exponentiation. Students can be shown that the successor operation is implicit in the "begats" of the Fifth Chapter of Genesis in the Bible. Adam begat Seth; Seth begat Enos; Enos begat Cainan; Cainan begat Mahaleel; Mahaleel begat Jared; Jared begat Enoch; Enoch begat Methusaleh. Thus, using a "begat operation", B(Adam) = Seth; B(B(Adam)) = B(Seth) = Enos; B(B(B(Adam))) = B(B(Seth)) = B(Enos) = Cainan; ...; B(B(B(B(B(B(B(Adam))))))) = ... = Methusaleh. Then, Methusaleh is the seventh generation from Adam, just as 7 is the seventh generation" from 0. The successor operation generatively explains the saying that "recursion is infinity in finite terms".
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