Proving or disproving a set of total functions is countable Let S be the set of total functions from $N \rightarrow M$, such that for each $f \in S$, there is $i &gt; 1$ such that for
  all $j &lt; i$, $f(i)$ 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.
Hint: Let $M_1,M_2$ be an arbitrary pair of inequivalent Turing machines. Your set $S$ contains all functions satisfying $f(1) = M_1$ and $f(2) = M_2$.