 a contradiction. Comparing this situation with  that described at the beginning of  Section 2.2  we can see that the partial computable functions differ from the primitive recursive functions in admitting a universal function within the same class but at the same time giving up the requirement that the functions in the class must be total. In other words, while \(\upsilon(i,x) \in \mathbf{PartREC}\), the discussion in  Section 2.2.2  shows that \(u_1(i,\vec{x}) \in \mathbf{REC} - \mathbf{PR}\). 