The recognizability of $\overline{A}_{TM}$

Question

In undergrad theory classes, the idea of decidability and recognizability is introduced. It's well known that $A_{TM}$, the set of words accepted by a TM $M$, is recognizable but not decidable. We also know that the complement problem $\overline{A}_{TM}$ is co-recognizable but not recognizable.

I have a doubt about the co-recognizability part. If $\overline{A}_{TM}$ is co-recognizable, then we have a TM that recognizes the set of words not accepted by some TM $M$. Doesn't that mean $\overline{A}_{TM}$ is recognizable, since we have a machine that recognizes the language?

I know this can't be the case, but I'm not able to find out where I'm going wrong with my logic.

Answer 1

This statement is incorrect.

If $\overline{A_{TM}}$ is co-recognizable, then we have a TM that recognizes the set of words not accepted by some TM $M$.

The fact that $\overline{A_{TM}}$ is co-recognizable means that there is a TM $T$ that recognizes all words not in $\overline{A_{TM}}$. Equivalently, $T$ recognizes all words in $\overline{\overline{A_{TM}}}=A_{TM}$, i.e., $T$ recognizes all words accepted by $M$.