If I model the natural numbers with some object $0$ and an injective computable function $s$ such that $ \forall n ( s(n) \ne 0 )$. Then due to Rogers' fixed-point theorem there must be a natural number $\Omega$ for which $ s(\Omega) = \Omega $.
Am I overlooking something obvious here?
The theorem only gives you a number $\Omega$ such that $\varphi_\Omega$ is the same partial recursive function as $\varphi_{s(\Omega)}$ (where $\varphi_n$ is the partial recursive function computed by, say, the $n^{\text{th}}$ Turing machine). It doesn't say that $\Omega=s(\Omega).$