Is this Language decidable?

Question

As the title says; is this language decidable and how do you prove it?

$$L =\{\langle M\rangle \mid M \text{ is a Turing Machine and there is an input that } M \text{ halts on} \} $$

Answer 1

Your language is not decidable. To show that it suffices to notice that the existence of a Turing machine $T$ that decided $L$ implies the existence of a Turing machine that solves the halting problem.

Indeed, given any Turing machine $M$ with input $x$ you can construct a Turing machine $M'$ that ignores its input, writes $x$ on the tape, and then simulates $M$. Clearly $M'$ halts (regardless of its input) if and only if $M$ halts on input $x$. We can decide whether $M'$ halts by checking whether $M' \in L$ using $T$ with input $M'$.