&#62; He kept saying, "But so then you add that new number to the list!"<p>I think if you get down to it, this reveals an interesting point.  I think it suggests that the logic behind the proof wasn't made clear enough, that it's not a constructive proof, it's one of the first sophisticated proofs by contradiction that a student sees.  Since you aren't disproving a concrete thing - you are showing that some hypothetical number doesn't appear on some hypothetical list - it can seem very unsatisfying.<p>Also, why is it so ridiculous to try to add a missing number to the list?  There's an intuition that you're fighting with a proof like this:
0: Create an empty list
1: Find a real number not on your list
2: Add it to the list
3: goto 1
Obviously this process never terminates, but I think you have to think pretty hard before that bothers you.  After all, what is fundamentally different if I replace "real" with "integer" or "rational"? (rhetorical)<p>Finally, students are comfortable enough talking about the "set of reals".  Showing that there's no such thing as the "list of reals" reveals that there must be some difference between a set and a list - but at the time this proof is presented, has that distinction been made sufficiently clear?