I should have never got drawn into this stupid thread.

Though talking with trainwreck was interesting and made me go back and read Cantor's diagnolization proof again since it's one of the coolest pieces of mathematics ever written. If you ever want to see an interesting development of the real numbers Baby Rudin is a really cool book, though challenging as since he writes everything as slickly as possible and doesn't use any diagrams. Plus the exercises are really hard. Terry Tao has some lecture notes he posted where he very carefully constructs the reals starting from only the Peano axioms for his real analysis class (wish he was teaching it when I took the class). Or it might be a textbook now.