@JohanWästlund Even if there are countably infinite mathematicians, a strategy exists such that at most one mathematician fails. See Can an infinite number of mathematicians guess the number in a box with only one error?

I think mathematician N cannot determine N since they don't know whether v[N]=M[N]. Therefore what we can actually say is "Mathematician i with i<N can determine N, however each mathematician with i≥N only knows that N≤i." The rest of the proof works anyway but it took me a while to see why; I have set out my reasoning in a new answer.