In order to prove a certain function to be partially computable, I need to show an $\mathbb S$-program that computes it. I could really use the predicate $X \in B$ in my program to draw my conclusion. To give you the idea of what I am dealing with here it is one of my problems:Give an infinite set $B$ such that $\Phi(x,x)\uparrow$ for all $b \inB$ and such that $$H(x) = \begin{cases}1 \text{ if }\Phi(x,x)\downarrow \\ 0 \text{ if } x \in B \\ \uparrow \text{otherwise}\end{cases}$$ show that $H(x)$ is partially computable.I am wondering if membership for infinite set is decidable and therefore can be used to write $\text{IF } X \in B$ such program. Am I allowed?Edit: the notation $\Phi(x,x)\uparrow$ means the function is undefined.
Is membership of x in an infinite set decidable?
computability;decision problem