| Your argument has some gaps. Here are the two basic problems: 1. f(k) is not defined in your notation, only f(k,k) is. My guess is that f(k) is supposed to represent the real number defined by the k^th row of the table. 2. Whatever f(k) is supposed to be in your notation, you have not shown that it has a limit as k approaches infinity, let alone that z = f. In fact, f(k) cannot possibly have a limit f. If it did, then by the definition of limit, all but finitely many numbers in [0,1] would have to be within epsilon of f, for any epsilon > 0. We could then show that all but finitely many numbers in [0,1] have the exact same first trillion digits of their decimal expansion. Obviously this is absurd. This is what always happens with Cantor diagonal argument critiques. The critique invariably ignores the logically rigorous argument of the CDA itself, sets up its own new version or extension of the diagonalization construction, and then makes whatever false assumptions are needed for the new construction reach a conclusion contrary to the CDA. It's like walking into a sturdy bridge with a few pebbles in your pocket, arranging the pebbles into a flimsy tower, knocking the tower over and declaring that you've destroyed the bridge. |