The existence of the real root can be debated by asking if one can possibly select a node that cannot be backtracked to the root in the infinite binary tree. If not, then infinite numbers don't exist. That is, N# or W don't exist. If so, where is the root? The middle is certainly disconnected by uncountable infinity to maintain the overall consistency. If the diagonal argument must work unconditionally in the finite, how many conditions must we consider? With the diagonal argument, it is clear, because we have known the height must be no greater than the width as a condition for a long, long time. However, for some reason, that all countable infinities are the same by Cantor's definition made it work. If you somehow want to believe that all countable infinities are the same while all uncountable infinities are not necessarily so, then there you go, you are a true Cantorist. Just please do not mess with Beth numbers and keep a distance with your Aleph numbers.
The thing is, which consistent mathematical theory works better to explain physical science and social science in which we apply mathematics? I am using computer science and Christian faith for infinity to take real place in the universe of discourse. When it comes to physics, we don't know if the universe has an end, all depending on our definition of the physical universe or the rate of expansion based on the Big Bang theory.
By the way, I stand by my own Ayin numbers, because P(N) is certainly as countable as Q and N. It is not fair to change arguments at will depending on the objects with which we are dealing.
