 Consider a term \(\tau\) corresponding to a “program” which determines the partial computable program with index \(\ulcorner \tau \urcorner\) (as described in  Section 2.2.2).  We say that such a program is self-reproducing if for all inputs \(x\), the computation of \(\tau\) on \(x\) outputs \(\ulcorner \tau \urcorner\). Since in order to construct \(\tau\) it would seem that we need to know \(\ulcorner \tau \urcorner\) in advance, it might appear that self-reproducing programs need not exist. Note, however, that transposed back into our official terminology, the existence of such a program is equivalent to the existence of a number \(n\) such that \(\phi_n(x) = n\). And this is guaranteed by applying  Corollary 3.2  to the function \(f(x,y) = x\).