There exist countably many enumerable disjoint and inseparable sets

2017-10-17 20:08:44

I want to formally prove that there are countably many disjoint enumerable sets such that any two of them cannot be separated by a decidable set.

I know that there exist a computable function $f$ that takes only values 0 and 1, and this can be used to define two sets (let's say $\mathcal{A}$ and $\mathcal{B}$, where $\mathcal{A}=\{x$ $s.t. f(x)=1\}$ and $\mathcal{B}=\{x$ $ s.t. f(x)=0\}$).

Could you please help me to extend this arguments to find (countably) many such sets?