What a waste of two months!
I know this sounds stupid, but after reading Everything and More and giving a presentation on the diagonal method, I felt a sudden burst of energy to write a refutation of the diagonal method.
Transfinite theory is a hideous scar on mathematics. The theory is essentially a dichotomy couched between two paradoxes. It starts with Galileo's Paradox to infer that the rationals and the integers are the same size. It then uses the diagonal method (a form of the liar's paradox) to suggest that the reals are a different size. This creates the denumerable/nondenumerable dichotomy.
I had been following the thread that the diagonal method doesn't create a dichotomy, but it actually shows that Bolzano's interpretation of the Galileo's paradox was incorrect. I did this by showing that Cantor's methodology implies that the set of namable numbers is both denuemrable and non-denumerable.
Everything was looking fine. Until I reread the thesis. Then it dawned on me that I was using the same garbage techniques of the German Idealists. I was trying to derive meaning from paradoxes. Suddenly, I could see a thousand different interpretations of the paradoxes. That is the whole friggin' point of building house of cards on paradoxes, as paradoxes allow you to conclude whatever you want.
In this refutation, I wanted to emphasize that if you pulled the denumerable/nondenumerable dichotomy, you would be left with a richer version of the infinite where there are multiple layers of infinite sets. Rich Theory is really based on the size of arbitrarily large sets. All it really does is say that different sets have different attributes. The Natural Numbers are different from the Whole Numbers which are different from the Integers, which are different from the Rationals, and so on.
The reason I detest transfinite theory is because it injects paradoxes into the foundation of mathematics. We can follow these absurd little chains of reasoning from our contemplation of infinite paradoxes, but they will all be aburd as paradoxes allow you conclude anything you desire.
The true refutation of the theory is simply to point out that it is based on paradoxes. Such a refutation falls on deaf ears because it the proliferation of paradoxes is the very reason that the theory is popular. People love paradoxes. Give mankind a big lie and they will be your friend for life.
I think Rich Theory still has some promising. Rich theory was really based on studying the relative size of arbitrarily large sets. Such sets are not subject to the paradoxes of the infinite. My interpretation of the infinite paradoxes was simply indicating that there is no reason to think that layering that occurs with arbitrarily large sets disappears when you take them to infinity.
My current thoughts are that the best way to refute transfinite theory is to show that the system paradoxical reasoning in transfinite theory has the same roots as Marxism, Fascism, Nazism and the other repressive regimes that made minced meat of civilization in the 20th century.
Marxism used similar combinations of paradoxes and dichotomies. For example, you often see the liar's paradox used to prove that Democracy is wrong. A democracy might elect a leader who intends to dissolve the democracy. Marx's dichotomy was between the capitalist and worker.
Regardless, I can't stand to spend another second on the stupid beast.