I've been randomly looking into this and... I don't believe Cantor's argument! I'm not great at maths, though, so it's very possible I'm just being dumb, but I think that the argument is flawed (and that the conclusion is too). Here's why: it's impossible to add one to x rank of each number on an infinite list without getting to a number of the list, because... the list is infinite! The result obtained through this cannot be unique, as infinity necessarily comprises everything. Thus, isn't the proof based on a misunderstanding of the concept of infinity?
And yes, it is very ballsy of me to attack something that apparently was proven a century ago, and I dare hope you're gonna prove me wrong or tell me I misunderstood something, O you knowledgeable people :P