It is an easy task to devise an encoding for a computing formalism (such as  Turing Machines) that has a function with fixed encoding. That is, the encoding is a numbering $\psi : \mathbb{N} \rightarrow \textbf{P}^{(1)}$ where $\textbf{P}^{(1)}$ is the set of partial unary computable $\mathbb{N} \rightarrow \mathbb{N}$ functions. By a function with fixed encoding we mean a function $f$ that has exactly one $c$ such that $\psi (c) = f$. It is also simple to have such a numbering be realizable, that is, its codes can be decoded and executed by a universal computer, and furthermore it is possible to directly provide a code $u$ for a universal computer.It seems we have all the ingredients to falsify Rice's Theorem; since there is only one code for $f$, deciding if a function belongs to $\{f\}$ is a simple matter of checking if the function's code is $c$. At this point we must stop and check the theorem's formal statement, in somewhere such as Wikipedia, where we find out that our numbering has to admissible. The proposed numbering is obviously not so; there is no computable function that could map exactly all standard codes for $f$ into exactly $c$.But that does not mean $\psi$ is an undesirable numbering. On the contrary, i'd argue that having a numbering where a function is restricted to a special code could be a more pleasant evironment to program in, and while i've only spoken of fixing an encoding for a single function, it should be generalizable to finite amounts of functions, and there might even be infinite classes as well (but that's a question for another TCS.se post). It's also clear that most properties over $\textbf{P}^{(1)}$ are still undecidable even under $\psi$, which suggests a more general restatement of Rice's Theorem.My question is: Are there generalizations of Rice's Theorem that account for realizable numberings, not just admissible ones?
Is there a formulation of Rice's Theorem that does not involve admissible (or Gdel) numberings?
computability