5.9 simpler solution

[Flecart]

Consider the machine that accepts everything. Clearly this machine is in the language T.

Now we mapping reduce from HALT to L using that machine. We need to prove that
$\forall \omega \in HALT \iff f(\omega) \in T$
Let's define

$HALT$ to be

$$ HALT = { \langle x, y \rangle \in \Sigma^{*} \times \Sigma^{*}: x = code(M),M \text{ halts on } x} $$

Now we set $f$ as follows:

if $x \neq code(\mathcal{M}) \forall \mathcal{M}$ then we return $x$, which clearly is not in $T$
Else we return the machine $N$ as defined:

1. With input $z$ ignore the input.
2. Simulate $x$ on input $y$.
3. If $x$ halts then accept the input.
4. Else continue to loop with $x$.

We see that $\langle x, y \rangle \in HALT \iff N \in T$

Now, if we can decide the language $T$ we would able to decide $HALT$, which is absurd.