Μ operator
Properties

(i) In context of the primitive recursive functions, where the search variable y of the μ-operator is bounded, e.g. y < z in the formula below, if the predicate R is primitive recursive (Kleene Proof #E p. 228), then

(ii) In the context of the (total) recursive functions, where the search variable y is unbounded but guaranteed to exist for all values x of the total recursive predicate R's parameters,

then the five primitive recursive operators plus the unbounded-but-total μ-operator give rise to what Kleene called the "general" recursive functions (i.e. total functions defined by the six recursion operators).

(iii) In the context of the partial recursive functions: Suppose that the relation R holds if and only if a partial recursive function converges to zero. And suppose that that partial recursive function converges (to something, not necessarily zero) whenever μ y R(y, x,..., x) is defined and y is μ y R(y, x,..., x) or smaller. Then the function μ y R(y, x,..., x) is also a partial recursive function.

The μ-operator is used in the characterization of the computable functions as the μ recursive functions.

In constructive mathematics, the unbounded search operator is related to Markov's principle.
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