Suppose we have the language
$ \overline L = \{x∈\{0,1\}*:x∉L\}$
And we know that
$A ≤ HPL ≤ \overline A$
where HPL is the Halting problem.
We need to state whether A is finite, countably infinite and whether $A$ and $ \overline A$ is recursive or not or re or not.
We know that A reduces to HPL and HPL is re and therefore A is re. Since HPL reduces to $\overline A$ and HPL is not recursive then $\overline A$ is not recursive. Furthermore, if we know that a language is not recursive but its complement is re then the language is not re. We know so far that $\overline A$ is not recurisve but $A$ is re then it means that $\overline A$ is not re. Furthermore if you know that a language L is not re then either $\overline L$ is not re or $\overline L$ is re but not recursive. We know that $\overline A$ is not re but A is re and therefore, we can say that A is not recursive.
Can anybody tell me if my deduction is any good? Also, how can I know if A is finite/ count infinite? Any tip would be greatly appreciated!