Georg Cantor’s diagonal proof is a thing of beauty.
A fine example of Murray Gell Mann’s noted concurrence of natural complexity and theoretical simplicity, the diagonal proof demonstrated with extreme brevity the uncountability of the real numbers, leading to the formulation of the aleph numbers and the variable cardinality of infinite sets. Ultimately this uncountability would inspire Cantor to posit his famous continuum hypothesis, establishing that the cardinality of infinite sets occurred only in discrete magnitudes, a hypothesis later proved undisprovable by Kurt Gödel and even later proved undisprovably unprovable (!) by Paul Cohen.
Lun-Yi Tsai beautifully rendered Cantor’s diagonal proof in this piece entitled Surprised Again on the Diagonal, presented in this month’s Tufts Magazine.
