Errors From Cantor to Hilbert

I will make a brief comment on the paradox called Hilbert's Hotel, since it has gotten a fair amount of attention recently. Hilbert imagines a hotel with infinite rooms. On one night, the hotel is found to be full. Hilbert then asks, "what if someone else comes to reception, begging a room?" What do we do? This paradox has been used to justify various types of transfinite math. Entire forests have been felled commenting on the mathematical implications of this paradox. However, it can be dismissed with one comment. If the infinite hotel is full, then no one can come begging a room. They are all already in the hotel. It is a contradiction in terms to imagine that you can add one to infinity in the first place. Where did the one come from? How is it possible for a quantity to be off an infinite number line? The simple and direct answer is that it is not possible. You cannot add one to infinity, because an infinite set is a complete set. An infinite set is complete in the fullest sense, meaning that there exists nothing outside the set. If you have an infinite set of people, then all people are in that set. You cannot postulate another person. Hilbert's Hotel is not a paradox, it is a very bad logical mistake, from the first paragraph. It is based on the same terrible mistake that underlies all transfinite math. The mistake is believing that the word "transfinite" can mean something. What it means in practice is really "transinfinite." Mathematicians believe that something can exist beyond infinity.
      If you accept the addition of 1 to infinity, then it means that you don't understand infinity to begin with. All the math that takes place in the transinfinite is quite simply false. Notice that I do not say it is physically baseless, or mystical, or avant garde, or any other half-way adjective. It is false. It is wrong. It is a horrible, terrible mistake, one that is very difficult to understand. It is further proof that Modern math and physics have followed the same path as Modern art and music and architecture. It can only be explained as a cultural pathology, one where self-proclaimed intellectuals exhibit the most transparent symptoms of rational negligence. They are outlandishly irrational, and do not care that they are. They are proud to be irrational. They believe—due to a misreading of Nietzsche perhaps—that irrationality is a cohort of creativity. Or it is a stand-in, a substitute. A paradox therefore becomes a distinction. A badge of courage. A brave acceptance of Nature's refusal to make sense (as Feynman might have put it.) If we somehow survive this cultural pathology, the future will look upon our time in horror and wonderment. How did we ever reach such fantastic levels of intellectual fakery and denial, especially in a century steeped in the warnings of Freud to beware of just this illness?

How is it possible for a quantity to be off an infinite number line? The simple and direct answer is that it is not possible. You cannot add one to infinity, because an infinite set is a complete set.

I don't understand at all what you're getting at, or what you mean by "complete".  Are you suggesting that we can't take the integers as an infinite set, because the set of naturals is "complete" and cannot be extended?
Can you take a power set of an infinite set?  That's transfinite math.  What about measure theory, or even simple integral calculus?  These disciplines rely on taking "measurements" of infinite sets, and have been proven effective when applied to the real world.  Is it just a coincidence that such manipulations of the infinite happen to apply to the real world?
You might want to look into finitism, a philosophy which I'm not too familiar with myself at the moment.
http://en.wikipedia.org/wiki/Finitism