Is there a numbering (not Gdel numbering) of all computable functions $U(p, x)$, such that the set of numbers of functions defined in zero is exactly the set of even numbers. More formally: $I = \{p,\ |\ U(p, 0)\  \mathrm{defined}\} = 2\mathbb{N}$.My guess that it's true. But I'm not sure how to prove it.Ideas:We can construct a numbering of all computable functions, defined in zero using the function $F(p, x, t)$ which is equal to $0$ if $U(p, x)$ hasn't finished work in $t$ steps and $1$ in other case. We can do it because set of pairs $(p, t)$ is enumerable.Then, having this function $V(p, x)$ and some other numbering $U(p, x)$ we can construct numbering$$U'(p, x) = \begin{cases}V(\frac{p}{2}, x)\ \ if\ p \vdots 2\\U(\frac{p + 1}{2}, x)\ \ if\ p \not\vdots 2\end{cases}$$
Numbering of computable functions
computability;discrete mathematics
Hint:There are countably many computable functions defined at zero, and countably many computable functions undefined at zero.There are countably many natural numbers divisible by 2, and countably many natural numbers not divisible by 2.Put the two facts together.