How to prove the union of languages recognized by a set of turing-recognizable Turing machines is also turing-recognizable?

Question

Let $G = \{\langle M_1\rangle, \langle M_2\rangle, \langle M_3\rangle,\cdots\}$ be an infinite turing recognizable language, whose members are descriptions of turing machines.

How can one prove that the union of all the languages recognized by the turing machines in $G$, $L(M_1) \cup L(M_2) \cup L(M_3)\cup\cdots $ is turing recognizable?

I tried to prove it by building an enumerator for the language, using the fact that since $G$ is turing recognizable it can be produced by one.

E' = 
1. Initialize steps counter i=0
2. Run E, the enumerator of G, number of steps equal to i.
3. For every turing machine description M produced by E:
   3.1 for every input word corresponding to 0 up to i according to lexicographic order do:
      3.1.1 Run M on the current input.
      3.1.2 print if M accepts.
4. increase i by 1, and go back to step 2.

Is this a correct approach? Running the enumerator for a limited amount of steps would produce the same words over and over again. Is this not a problem when creating an enumerator? Is $E'$ valid and producing the language as needed? If not, what should be changed?

Answer 1

Your enumerator is incorrect. Because there are infinitely many Turing machine descriptions in $G$, the loop corresponding to Line 3 will not halt, so i will never be increased. Also, Line 3.2 may not halt too.

The correct enumerator is:

E' = 
for i = 1 to infinity:
    Run E, the enumerator of G, to get the ith description of Turing machine <Mi>
    Build an enumerator Ei for Mi 
    for j = 1 to i - 1:
        Run Ej to get the ith string, print it
    Run Ei to get the first i strings, print them

Suppose $E_i$ prints $s_{i1},s_{i2},\ldots$ in order. Then after $i$ iterations, $E'$ would have printed $s_{11},\ldots,s_{1i},s_{21},\ldots,s_{2i},\ldots,s_{i1},\ldots,s_{ii}$ (note this is not the order they are printed), and would print $s_{1(i+1)},s_{2(i+1)},\ldots,s_{i(i+1)},s_{(i+1)1},\ldots,s_{(i+1)(i+1)}$ in order in the $(i+1)$th iteration. Of course, before printing a string, you can check if it has been printed to avoid duplicates.