The aleph

Two shepherds want to know which of them owns the larger flock. Neither knows how to count — no matter. They let their animals pass two by two, one from each flock, and watch which flock runs out first. Pairing is enough: the comparison of sizes is there before the numerals.

In 1874, Georg Cantor took this shepherd's gesture seriously and carried it where no one expected it: to infinite collections. Two sets have the same size, he decreed, when their elements can be paired off with nothing left over — each with exactly one counterpart. The definition looks harmless. It is a thunderclap.

For the even numbers pair off with all the integers: 1 with 2, 2 with 4, 3 with 6, and so on, with no one left out. The part is as large as the whole. Galileo had stumbled on the same scandal with the perfect squares, three centuries earlier, and had cautiously concluded that “greater” and “smaller” had no currency in the infinite. Cantor concluded the opposite: they do, provided one counts by pairing. Even the fractions, so dense that between any two of them a third always fits, can be lined up in a single file: ℚ is no larger than ℕ.

One then believes one has the rule — all infinities would be equal. It is here that Cantor produced one of the most economical arguments in mathematics. Suppose the complete list of the real numbers between 0 and 1 has been drawn up: the first, the second, the third, each with its decimals. Now build a number by changing the first decimal of the first, the second of the second, the n-th of the n-th. This number differs from every line of the list — from the n-th precisely at the n-th decimal. So it appears nowhere on it. The list that was called complete was not, and none can be: the continuum overflows every enumeration.

There are, then, infinities larger than others. To the smallest — that of the integers — Cantor gave a name that belonged to no mathematical alphabet yet: ℵ₀, aleph-zero, first letter of the Hebrew alphabet. Above it come ℵ₁, then ℵ₂ — a ladder with no last rung. Whether the continuum occupies the first floor above the integers is the continuum hypothesis; Gödel and then Cohen showed that the ordinary axioms of mathematics do not decide it. The question outgrew the house that was meant to lodge it.

Readers of the house know this letter: it is the golden aleph on the covers of the Articulated Course in Mathematics. Borges made of it the point where the entire universe holds in a single place; the Semitic languages saw in it the head of an ox. To count is to pair — and the infinite begins with the shepherd's gesture.

The Articulated Course in Mathematics spans more than 5,600 pages: you do not enter it at random. So the house has published its first special issue — a thirteen-page Reading Guide that gives the door according to where you come from and the path according to where you are going: seven portraits of readers, the map of what each volume assumes and what it builds, three tracks that cross the collection without reading everything, and reading plans over twelve, twenty-four and fifty-two weeks.

It is made exactly like the volumes: the same 105 × 170 mm pocket format, the same Garamond, the same cream paper — and, on the cover, the same golden mosaic: the aleph of the note above. It is the house's workshop applied to its smallest object.

It is free, requires no sign-up, and may be copied and shared freely, provided it is shared whole and unchanged. Download the Reading Guide — or read it online at clementinium.com/guide.

This is a first issue: nothing has changed yet, and that is precisely what this section records. The five volumes of the Course and the essays are at version 1.0; no correction is on the register to date.

The house rule fits in one sentence: every reported error is a corrected error. Corrections are recorded in the public register (clementinium.com/errata), with exact page and statement; when they warrant a corrected edition, the files are remade, and every buyer re-downloads the latest version from their library (clementinium.com/bibliotheque) — free, for life.

Found an error? The register awaits its first entry. Report it on the register page, or to [email protected]: issue No. 2 will record it.

This summer the Reading room opened (clementinium.com/reader): a continuous excerpt of each of the five volumes can be leafed through online, with table of contents and search, no sign-up required. It will keep growing.

Three works are under way. Printed editions are being tried — paper proofs, in the same pocket format, are in preparation; the collection will only appear in print if the making is worthy of it. A companion of hints and solutions for Volume I is under study: answers to routine exercises, graduated hints for the important problems, full solutions for a few representative ones. And a general index of the catalogue — definitions, theorems and notation, across the volumes — is in the works.

Issue No. 2 will appear in the autumn.