Recursively enumerable language of Turing machines

Question

If you have the language $L_{h}=\{M_{i} | (\exists z \in \sum ^{*}) M_{i}\text{ halts on some input } z\}$

where $M_{i}$ are Turing machines, is $L_{h}$ recursively enumerable?

I'm fairly certain it is, but I'm having issues proving that to be the case. The way I understand RE languages is that they CAN be infinite, which this language most certainly is, but I have no idea where to go from there.

Thanks in advance.

Answer 1

Hint:

• First, is the language of all Turing machines enumerable?
• Consider a language $L_i = \{M_i \mid M_i \text{ halts on some input }\}$. Its size is at most $1$, how to "enumerate it"?
• Run in parallel $L_i$ for all Turing machines, and if some of $L_i$ finishes, output its unique element.

I hope this helps $\ddot\smile$