# Is the complement of every non Turing recognizable language a Turing recognizable language?

One thing I understand is that the complement of every Turing recognizable(but not decidable) language is non Turing recognizable.

What about the complement of non Turing recognizable language? Is the complement of every non r.e a recognizable language.

I don't know how to approach this problem.

In fact, the language of all total Turing machines is $\Pi_2$-complete, which means, in a sense, that there is no better way to "solve" it than to run all on inputs and see whether the given machine halts on each of them (of course, you can't do this with a Turing machine, but perhaps with a more powerful device...). Take a look at the arithmetical hierarchy for more on this.