That's a good sketch of some of the complexities involved. I remember that some months after I learned the proof, your rhetorical question 'What is fundamentally different if I replace "real" with "integer" or "rational"?' occurred to me, and I had to spend a few weeks thinking about it before the answer came to me.<p>In other words, I got to have an experience of actual mathematical thinking without learning much higher math. This is why I love the diagonal proof, and the guy who taught it to me: Paul Lockhart, author of the essay 'A Mathematician's Lament.' (<a href="http://www.maa.org/devlin/devlin_03_08.html" rel="nofollow">http://www.maa.org/devlin/devlin_03_08.html</a>)