Immutable Antinomy · Track 17 · middle
Ode to Galileo's Paradox
There are as many perfect squares (1, 4, 9, 16...) as there are positive integers, even though squares are a subset of integers. Galileo noticed and was troubled. Cantor later resolved it — and opened infinity into levels. An ode to the counterintuitive arithmetic of the infinite: a part can equal the whole.
Lyrics
[Intro] Arcetri. The winter of 1637. A high-backed chair, a coal brazier spitting dust. My scribe wears gloves against the chill. I am speaking through Salviati now. [Verse 1] Take the numbers. Just the counting ones. One, two, three, four. A line stretching out of sight. Now, take their squares. One, four, nine, sixteen. For every number, there is a square. For every square, there is a root. A perfect pairing, one for one. But the squares are just a part. They leave gaps. Two, three, five, six, seven, eight. How can the part be as long as the whole? [Chorus] There are as many squares as there are numbers. The subset is as vast as the set. The part equals the whole. And I do not see how to escape it. The words 'greater' and 'less' have broken. They do not apply to the infinite. They die right here, in this cold room. [Verse 2] In Leiden, a year from now, the press will run. Elsevirs' lampblack ink on damp rag paper. A small book, an octavo, 280 grams of trouble. They will read my words, Salviati's words. And they will feel the floor give way. Just as I did. This perfect mapping, this terrible correspondence. Proving that the container is no larger than what it contains. [Chorus] There are as many squares as there are numbers. The subset is as vast as the set. The part equals the whole. And I do not see how to escape it. The words 'greater' and 'less' have broken. They do not apply to the infinite. They die right here, in this cold room. [Bridge] Two hundred and thirty-six years pass. Georg Cantor picks up my fallen thread. He is not troubled. He is freed. What I saw as a paradox, he sees as a door. He gives my broken rule a name. Aleph-null. The first step on a ladder of infinities. He proves there are more numbers between zero and one than all the numbers I ever counted. My little paradox was just the beginning. [Outro] But it began here. With an old man, going blind, dictating to a boy. The coals in the brazier are turning grey. The scribe's hand is cramping. One to one. Four to two. Nine to three.