I shared this observation with my eldest son, Joe, early in his college career, when he told me he’d declared math as a major. Chesterton’s warning wasn’t against using logic, only embracing it to the exclusion of all else. The topic came up again recently over dinner when Joe, now a senior, explained something counterintuitive to his old man.
He said that between integers—say, 1 and 2—there are infinitely many real numbers, like 1.1 and 1.265. Such thinking scared me straight into law school at his age, yet somehow I grasped it now, if only conceptually. “Like the Incarnation,” I offered. An instance of the Creator, while remaining fully God and fully man, entering into his creation: the infinite bounded by the finite.
“I suppose,” Joe answered, checking my catechism against his set theory. Then he said something even trippier. Although whole numbers can be listed out to infinity, the hypothetical list of real numbers is necessarily larger than the hypothetical list of whole numbers. Not all infinities are equal.