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, etc. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as synonym for number theory. ... Addition is one of the basic operations of arithmetic. ... In its simplest form, multiplication is a quick way of adding identical numbers. ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...

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, 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. For closure in computer science, see closure (computer science). ...

Students can be shown that the successor operation is implicit in the "begats" of the Fifth Chapter of Genesis in the Bible. The Bible (From Greek (τα) βιβλια, (ta) biblia, (the) books, plural of βιβλιον, biblion, book, originally a diminutive of βιβλος, biblos, which in turn is derived from βυβλος—byblos, meaning papyrus, from the ancient Phoenician city of Byblos which exported this writing material), is a word applied to sacred scriptures. ...

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".