Successor function

A 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.

Definition: S(n) = n + 1, for initial input 0.

Thus, S(0) = 0 + 1 = 1; SS(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 + S(b) = a x b = a.

Exponentiation: b⁰ = 1, b^S(e) = (b^ez) 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".