so-called minimization- or mu-operator: Defined as "the least value of y less than z such that R(x, y) is true; or z if there is no such value." #F: Definition by cases: The function defined thus, where Q1, ..., Qm are mutually exclusive predicates (or "ψ(x) shall have the value given by the first clause that applies), is primitive recursive in φ1, ..., Q1, ... Qm: φ(x) = φ1(x) if Q1(x) is true, . . . . . . . . . . . . . . . . . . . φm(x) if Qm(x) is true φm+1(x) otherwise #G: If φ satisfies the equation: φ(y,x) = χ(y, COURSE-φ(y; x2, ... xn ), x2, ... xn then φ is primitive recursive in χ. The value COURSE-φ(y; x2 to n ) of the course-of-values function encodes the sequence of values φ(0,x2 to n), ..., φ(y-1,x2 to n) of the original function. Use in first-order Peano arithmetic In first-order Peano arithmetic, there are infinitely many variables (0-ary symbols) but no k-ary non-logical symbols with k>0 other than S, +, *, and ≤. Thus in order to define primitive recursive functions one has to use the following trick by Gödel. By using a Gödel numbering for sequences, for example Gödel's β function, any finite sequence of numbers can be encoded by a single number. Such a number can therefore represent the primitive recursive function until a given n. Let h be a 1-ary primitive recursion function defined by: where C is a constant and g is an already defined function. Using Gödel's β function, for any sequence of natural numbers (k0, k1, ..., kn), there are natural numbers b and c such that, for every i ≤ n, β(b, c, i) = ki. We may thus use the following formula to define h; more precisely, m=h(n) is a shorthand for the following: and the equating to g, being already defined, is in fact shorthand for some other already defined formula (as is β, whose formula is given here). The generalization to any k-ary primitive recursion function is trivial. Relationship to recursive functions The broader class of partial recursive functions is defined by introducing an unbounded search operator. The use of this operator may result in a partial function, that is, a relation with at most one value for each argument, but does not necessarily have any value for any argument (see domain). An