Just as our faith in Cantor grew in the last post, doubt is once again starting to creep in. I'm talking about the well-ordering principle, which says, unsurprisingly enough, that any set can be well-ordered. What this means for the reals is that there is a total ordering relation (which most emphatically is not the usual "less than" relation on R) -- let's denote it by x <' y -- that well-orders the reals. This relation induces the ordering
a <' b <' c <' ...
and every real number r will eventually appear in this chain.
Now the $1.64 question is: why can't we apply Cantor's diagonal argument to this well-ordering to construct a number that doesn't appear in the chain?