# True or False: If $A \subseteq \{0,1\}^* \Rightarrow A^*$ is semi-decidable

Question: Is the following statement true or false?

If $A \subseteq \{0,1\}^* \Rightarrow A^*$ is semi-decidable

I thought that since every language is automatically of type 0, it follows that $A \in RE$, and since RE is closed under the Kleene star operator, it follows that $A^* \in RE$ and therefore semi-decidable.

However the solution says it's not true and uses the following counterexample:

If $A = \{1^n0 | 1^n \not\in H'\}$, where H' is the unaryly coded halting problem. Since $\overline{H} \leq A$, A is not semi-decidable. If $A^*$ were semi-decidable then so would be $A^* \cap \{1\}^* = A$, since $\{1\}^*0$ is semi-decidable and RE is closed under intersection, which is a contradiction.

Since I don't fully understand that line of reasoning, could someone please tell me what's wrong with my argumentation?