5.4 The Dedekind-Peano Axioms
For those that are math-averse, don't let this frighten you -- this will be a peaceful journey that should not leave you bewildered. Rather, it will provide some nice background for how recursion came to be used. With the history reviewed, we'll make our way into practical implementations.
Despite the fluorescence of maths in the 17th and 18th centuries and the growing impact of number theory, the ground upon which mathematics were built was shaky at best. Indeed, what we now consider to be the foundations of mathematics had not even been agreed upon (and this didn't happen until the first half of the 20th century with the maturation of logic and rise of axiomatic set theory).
One of the big problems facing mathematicians and one that also prevented the clarification of the foundations, was this: a thorough, precise, and consistent definition of the natural numbers as well as operations that could be performed on them (e.g., addition, multiplication, etc.). There was a long-accepted intuitive understanding, however, this was insufficient for complete mathematical rigor.
Richard Dedekind addressed this with his method of cuts, but it was Giuseppe Peano that supplied us with the clearest, most easily described axioms defining the natural numbers and arithmetic, wherein he made effective use of recursion. His definitions can be easily found in text books and on the Internet; we will take a slightly unique approach, however, and cast them in terms of LFE.
5.4.2 A Constant and Equality
The first five Peano axioms deal with the constant (often written as "0") and the reflexive, symmetric, transitive and closed equality relations. These don't relate recursion directly, so we're going to skip them ;-)
5.4.3 Successor Function
The concept of the "successor" in the Peano axioms is a primitive; it is taken as being true without having been proved. It is informally defined as being the next number following a given number "n".
The things to keep in mind here are that 1) we haven't defined addition yet, and 2) you must not interpret "+" as addition in this context, rather as the operator that allows for succession to occur. In the world of the Peano axioms, "+" is only validly used with "n" and "1".
This function is defined as being "basic primitive recursive". The basic primitive recursives are defined by axioms; the term was coined by Rózsa Péter.
5.4.4 The Remaining Axioms
The remaining three Peano axioms do not touch upon recursion directly, so we leave them to your own research and reading pleasure.