While some textbooks use the μ-operator as defined here,Enderton, H. B., A Mathematical Introduction to Logic, Academic Press, 1972Boolos, G. S., Burgess, J. P., Jeffrey, R. C., Computability and Logic, Cambridge University Press, 2007 othersJones, N. D., Computability and Complexity: From a Programming Perspective, The MIT Press, Cambridge, Massachusetts, London, England, 1997Kfoury, A. J., R. N. Moll, and M. A. Arbib, A Programming Approach to Computability, 2nd ed., Springer-Verlag, Berlin, Heidelberg, New York, 1982 demand that the μ-operator is applied to total functions only. Although this restricts the μ-operator as compared to the definition given here, the class of μ-recursive functions remains the same, which follows from Kleene's Normal Form Theorem (see below). The only difference is, that it becomes undecidable whether a specific function definition defines a μ-recursive function, as it is undecidable whether a computable (i.e. μ-recursive) function is total.
While some textbooks use the μ-operator as defined here,Enderton, H. B., A Mathematical Introduction to Logic, Academic Press, 1972Boolos, G. S., Burgess, J. P., Jeffrey, R. C., Computability and Logic, Cambridge University Press, 2007 othersJones, N. D., Computability and Complexity: From a Programming Perspective, The MIT Press, Cambridge, Massachusetts, London, England, 1997Kfoury, A. J., R. N. Moll, and M. A. Arbib, A Programming Approach to Computability, 2nd ed., Springer-Verlag, Berlin, Heidelberg, New York, 1982 demand that the μ-operator is applied to total functions only. Although this restricts the μ-operator as compared to the definition given here, the class of μ-recursive functions remains the same, which follows from Kleene's Normal Form Theorem (see below). The only difference is, that it becomes undecidable whether a specific function definition defines a μ-recursive function, as it is undecidable whether a computable (i.e. μ-recursive) function is total.