How to prove the language of all Turing Machines that accept an undecidable language is undecidable?

Question

I want to prove that $L=\{\langle M \rangle |L(M)\text{ is undecidable}\}$ is undecidable

I am not sure about this. This is my try :

Suppose L is decidable. Let $E$ be the decider from $L$. Let $A$ be a TM which is recognizing $A_{TM}$. Let $S$ be a TM which works on input $\langle M,w \rangle$ in the following way(the goal is that $S$ will be a decider for $A_{TM}$):

1. Construct a TM $N$ which works on Input $x$ as follows: Run $M$ on $w$. If $M$ $accepts$ run $A$ on $x$ and accept $x$ if $A$ accepts.(In this case is $L(N)=A_{TM}$). If $M$ $rejects$ $w$, $accept$ $x$.(In this case is $L(N)=\Sigma^*$)
2. Run $E$ on $N$ and accept if N accepts. Otherwise reject

I am not sure if my reduction is the right way or not. Maybe someone can help to finish the reduction :)

Answer 1

I think you are on the right track but things need to be made more explicit. First, what machine exactly is $A$? Any machine? Where does your contradiction appear?

First, you need to explicitly say that $A$ is a machine for an undecidable language, for example, let $A$ be the universal machine, that on inputs $\langle M \rangle w$ simulates $M$ on $w$, and accepts if $M$ accepts $w$.

Then, say explicitly that $S$ is a decider (a machine that always halts), and include the simple extra steps that show that $S$ accepts $\langle M \rangle w$ if and only if $w \in L(M)$. Finally, recall that $S$ is then a decider for a problem we already know is undecidable; a contradiction.

Edit: Sorry, it seems that $A_{TM}$ is standard notation for the universal machine. Forget about that comment if that is the notation used in your context.