There are infinitely many recursively enumerable subsets of the natural numbers which are not recursive

Question

How do I prove this claim? I understand that there are countably many recursive as well as recursively enumerable sets, and that the natural numbers have uncountably many subsets.

Answer 1

If you have one recursively enumerable but not recursive set, you can turn it into infinitely many. Let the recursively enumerable but not recursive set be $A$. Let $C=\{2n+1|n \in A\}$. It is also recursively enumerable but not recursive. To enumerate it, just enumerate $A$ and as each element comes out, double it and add one. If it were recursive, so would $A$ be. Now define the family $B_i$ as $B_i=C \cup \{2j|0 \lt j \le i\}$