Intuitively, a recursive function is a positive integer valued function of one or more positive integer arguments which may be computed by a definite algorithm.

Recursive functions may be defined more rigorously as the largest class of partial functions from satisfying the following six criteria:

1. The constant function defined by for all is a recursive function.
2. The addition function and the multiplication function are recursive function.
3. The projection functions with defined as are recursive functions.
4. (Closure under composition) If is a recursive function and with are recursive functions, then , defined by is a recursive function.
5. (Closure under primitive recursion)If and are recursive function, then , defined by the recursion
   
   with the initial condition
   
   is a recursive function.
6. (Closure under minimization) If is a recursive function then is a recursive function, where is defined to equal if there exists a such that
   • are all defined,
   • when , and
   • , otherwise is undefined.

The operation whereby was constructed from and in criterion 5 is known as primitive recursion. The operation described in criterion 6 is known as minimization. That is to say, for any given function , the partial function constructed as in criterion 6 is known as the minimization of and is denoted by .

The smallest set of functions satisfying criteria 1-5, but not criterion 6, is known as the set of primitive recursive functions.

With some work, it can be shown that the class of recursive functions can be characterized by considerably weaker sets of criteria than those given above. See the entry “alternative characterizations of recursive functions” for several such characterizations.