A recursive form requires a unique initiator and (sometimes) a unique promoter, as well as a unique recursor to link initiator with promoter or to represent such an implicit linkage.

In a file on another of my websites, an entire "arithmetic" curriculum of 7+ stages begins by recursively defining natural (counting) numbers using as initiator, 0 (zero); as promoter, 1 (unit); and as recursor, S (successor or successor function). Then, using only the initiator and the recursor, S(0) = 1, S(1) = 2, S(2) = 3, etc., without end! Using also the promoter, S(0) = 0 + 1 = 1; S(1) = 1 + 1 = 2; S(2) = 2 + 1 = 3; etc. Thus, in finite terms, an infinitude is articulated. Generation-by-recursion provides primary arithmetic operations (addition, etc.). Other generative procedures provide inversive operations (subtraction, etc.). And explain the various number systems of arithmetic (natural number, integer, rational, real number, complex number). Savings accounts accumulate compound interest recursively. Money present accumulates interest, to be added to this amount, feeding back to earn new interest. In general, recursive output is input transformed into new output, becoming new input. ("Gee! Like sssssexxxxxxxxxx!") Roughly, in a recursive sentence, some words become sentences within this sentence.

In the language of "AFRICAN VIOLET ED" (on this website), I teach the conek of recursion to preschoolers with the loconeks of fingers or blocks; to school children via the loconeks of 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. (And, as Sportin' Life sings, in the Gershwins' Porgy and Bess, "Methusaleh lived nine hundred years. But who calls that livin when no gal will give in to no man who lived nine hundred years?") Take B(_) as the begat-function. Thus, B(Adam) = Seth; B(Seth) = Enos, or B(B(Adam)) = Enos. Then, B(B(B(B(B(B(B(Adam))))))) = Methusaleh, the 7th generation begat from Adam.

(E. Y. Harburg and Burton Lane wrote a "Begin the Begat" song for Finian's Rainbow, announcing, "They begat the misbegotten GOP!")

I modeled the begat-function after the arithmetician's successor-function, S(_), where S(n) = n + 1 -- the successor of number n is number n and one more. Thus, S(0) = 1; S(1) = 2, or (embedded) S(S(0)) = 2. Thus, S(S(S(S(S(S(S(0))))))) = 7. The 7th successor of 0 is 7, just as Methusaleh is the 7th begat of Adam.

Now recursive generation explodes.

Instead of operations by axiomatic "stork", addition is recursively generated from the successor function; multiplication is recursively generated from addition; exponentiation is recursively generated from multiplication. All primary arithmetic operations are (like language!) recursively generated. ("Word bone connected to the number bone!" American linguist, Leonard Bloomfield, said, "Mathematics is language at its best.")

(You weren't taught this? Neither was I -- even after 8 years and 103 credit hours in mathematics at Columbia U. and New York U. I gleaned this from mathematical literature.)

As to hiconeks, mathematics abounds with recursive definitions of powerful and complex concepts and procedures. (Some of these simpler recursive definitions are applied in my development of Arithmetic.) And, starting with the loconek of fingers, recursion becomes very hiconek in the advanced mathematical field of recursive function theory, which (among other "goodies") tells computer science experts which problems are solvable and which problems are not solvable. A file on ALGORITHMS, on this website, shows how the extension of RECURSION to RECURSIVE SET THEORY provides powerful extension of THE ALGORITHM, and the power of the COMPUTER.

As to a nonmathematical hiconek for recursion, MIT mathematical linguist, Noam Chomsky, has produced evidence that the unlimited extension of a language such as English is possible only by the recursive device of embedding sentences in sentences (a refinement of patterns in patterns).

Thus, a talky little girl may say, "Dorothy, who met the wicked Witch of the West in Munchkin Land where her wicked Witch sister was killed, liquidated her with a pail of water." Clearly, two simple sentences -- "Dorothy met the Wicked Witch of the West in Munchkin Land", and "Her sister was killed in Munchkin Land" -- can be embedded in the simple sentence, "Dorothy liquidated her with a pail of water", to obtain that very talky sentence. This embedding is the powerful generation by recursion I'm raving about.

Chomsky was concerned with rebutting claims that we learn language by trial-and-error associations, modeling his rebuttal upon the advanced mathematical theory of Markhov chains. The subject and predicate of "Dorothy liquidated her with a pail of water" are separated (in that talky sentence) by an 18-word clause, which is equivalent to an 18-order Markhov chain, equivalent to learning nearly 69 billion associations! (The child should live so long!) Thus, compressing approximately 69 billion associations into a 26-word sentence shows the power of recursion to embed the infinite in the finite.

Actually, recursion is required in even simpler cases than that talky sentence of that talky little girl. Steven Pinker (a colleague of Chomsky) notes, in The Language Instinct (p. 368), that even the possessive form in English requires recursion. Pinker gives as example, "the man's hat". The long form is "the hat of the man". But the possessive-apostrophe enables us to embed the "hat" term below the "man" term in an equivalent phrase.

(Please note that, informally, language provides a loconek for recursion. And the use of recursion to generate language and arithmetic -- "Letter- bone connected to the word-bone!" -- implies that generatic teaching of arithmetic, as described above, unites "The Three R's", whereas the prevailing teaching of arithmetic by quasi-axiomatic methodology fragments them!)

As a further hiconek of recursion, the 1984 Nobelist in Medicine and Physiology, Niels K. Jerne, used Chomsky's model to explain the human immune system, equating "components of a generative grammar ... with various features of protein structures". The title of Jerne's Stockholm Nobel lecture was "The Generative Grammar of the Immune System".