Cantor’s arguments always receive a very negative reception from most of the HN crowd, just as they did from his peers at the time! He was nearly shunned for opening this can of worms, which is one of the reasons why he spent many years refining his arguments to be as elegant as possible (diagonalization came long after the original result).

Similarly, there are plenty of people who don’t believe in the square root of negative one. Others don’t believe in the existence of irrational numbers (e.g. the Greeks). Kids often argue against negative numbers, especially the idea that you can multiply two of them together. Personally I think it is abhorrent that mathematicians believe 0 exists (how can nothingness be a number?!)

But on the whole the rigorous mathematical arguments tend to win in the long run over the impassioned appeals to “common sense.” Today Cantor is regarded as a hero by nearly all mathematicians.