In this paper, we construct a computable family \(\mathcal {R}\) of r.e. sets whose every computable numbering \(\alpha \) is complete and encodes the Gödel numbering \(x\mapsto W_x\) of the family of all r.e. sets within itself in the sense that there exists a recursive function r such that for every \(b\in \mathbb N\) there is a \(B\subseteq \mathbb N\) with \(\alpha (r(b))=B\oplus W_b\) . Then we prove that, for all \(n\geqslant 2\) , every non-trivial \(\Sigma ^0_n\) -computable family has a non-complete (and even non-cylindrical) \(\Sigma ^0_n\) -computable numbering, but there exists a \(\Sigma ^0_n\) -computable family \(\mathcal {A}\) whose every \(\Sigma ^0_n\) -computable numbering \(\beta \) has the fixed point property (i.e., for every recursive function f there is a \(p\in \mathbb N\) with \(\beta (f(p))=\beta (p)\) ) and encodes within itself the numbering \(x\mapsto W^{\emptyset ^{(n-1)}}_x\) .
