Let S be the set of total functions from $N \rightarrow M$, such that for each $f \in S$, there is $i > 1$ such that for all $j < i$, $f(i)$ and $f(j)$ are not equivalent Turing machines. Prove or disprove that S is countable
I don't think this is actually countable, but I'm having a hard tie proving this using diagonilzation. Can anyone give me pointers / hint? Preferably explained at a lower level.