Intuitve Proof that the halting set is recursively enumerable

Question

I am trying to understand intuitively how the halting set $K := \{(i, x) \mid$ program $i$ halts when run on input $x\}$ is recursively enumerable.

Is there a simple explanation which says why this is? I'm not looking for a proof or anything that contains too many technical terms. One way that I was told to think about it was if you plot Turing machines on the $x$-axis and time on the $y$-axis, you can sort of think about in a diagonalization proof sort of way. I might have missed something and I want to fully understand this way of thinking.

Answer 1

Consider the program which takes a pair $(i,x)$ and runs program $i$ on input $x$. The set of pairs $(i,x)$ for which this program halts is precisely $K$, so $K$ is recursively enumerable.

Answer 2

The key idea is that we can recursively simulate steps in the operation of a Turing machine program on a given input. So to list out the elements of $K$ first we can look at the source code for program 1 and simulate 1 step of its operation on input $x$. Next we can also look at the source code for program 2 and simulate 2 steps of operation of both programs 1 and 2 on input $x$. And so on, at stage $n$ simulating $n$ steps of operation of each of programs $1\ldots n$ on input $x$. At any stage if a program halts on the input, we spit it out in the list.