So I was trying to solve following exercise:

Let $ (L_{i})_{i \in \mathbb{N}}$ be a family of decidable languages – this means that every $ L_{i}$ is decidable. Then $ \cup_{i \in \mathbb{N}}L_{i} $ is also decidable. Right or wrong?

The solution states, that this statement is wrong because we can set $ L_{i}:=\{f(i)\}$ with $ f : \mathbb{N} \rightarrow H_{0}$ and will therefore receive $ \cup_{i \in \mathbb{N}}L_{i} = H_{0}$ . At the same time $ L_{i}$ languages are still decidable, for every $ i \in \mathbb{N}$ because they are finite.

However I still struggle to understand how exactly a language can be decidable with $ f : \mathbb{N} \rightarrow H_{0}$ , because I thought that $ H_{0}$ was not decidable.

Thanks in advance!!