One of the most profound moments in my life was when I first understood Cantor’s diagonal proof. The idea that some infinities can be larger than others is mind-boggling and yet Cantor found a way to actually prove it! Just let that idea sink in for a minute. An infinite set is larger than anything we can think of. Think of any number and add one to it.. at heart that is what an infinite set is (at least so I thought). And then, Cantor threw in a proof showing that some infinite sets are larger than others.
What does that even mean?
But the proof is so simple and straight-forward that one cannot deny its power. For reasons of space I will not not get into the proof, there are many available online (such as this one). But this is my typographical interpretation of that proof, where the countable numbers are written in rows (0.countablecountable…) and across the diagonal is a number that does not exist in the original list (0.uncountableuncountable… ). This image of course does no justice to the beauty of the original proof which is something each of us should savor individually.
Click image to enlarge (opens in a new window).