Editing Recursion (section)

===Recursively defined sets===
{{Main|Recursive definition}}

====Example: the natural numbers====
{{See also|Closure (mathematics)}}
The canonical example of a recursively defined set is given by the [[natural numbers]]:

:0 is in <math>\mathbb{N}</math>
:if ''n'' is in <math>\mathbb{N}</math>, then ''n'' + 1 is in <math>\mathbb{N}</math>
:The set of natural numbers is the smallest set satisfying the previous two properties.

In mathematical logic, the [[Peano axioms]] (or Peano postulates or Dedekind–Peano axioms), are axioms for the natural numbers presented in the 19th century by the German mathematician [[Richard Dedekind]] and by the Italian mathematician [[Giuseppe Peano]]. The Peano Axioms define the natural numbers referring to a recursive successor function and addition and multiplication as recursive functions.

====Example: Proof procedure ====
Another interesting example is the set of all "provable" propositions in an [[axiomatic system]] that are defined in terms of a [[proof procedure]] which is inductively (or recursively) defined as follows:

*If a proposition is an axiom, it is a provable proposition.
*If a proposition can be derived from true reachable propositions by means of inference rules, it is a provable proposition.
*The set of provable propositions is the smallest set of propositions satisfying these conditions.