1895 — ALEPH-NULL

Georg Cantor published "Beiträge zur Begründung der transfiniten Mengenlehre" in Mathematische Annalen, volume 46, in 1895. Section 6 of that paper is headed "the smallest transfinite cardinal number aleph-zero", and it is where the symbol ℵ₀ appears in print for the first time. Cantor had settled on the Hebrew letter two years earlier in correspondence — the Latin and Greek alphabets, he wrote, were already over-used — but 1895 is the year it went public, and no earlier printed use of it exists.

The subscript is the entire point. ℵ₀ is not a name for infinity; it is the size of one particular infinite set, the natural numbers, and it is the smallest infinite size there is. Anything you could list, even in principle, has exactly ℵ₀ elements: the even numbers, the primes, the integers including the negative ones, every fraction that can be written. All of them can be matched one-to-one with 1, 2, 3, and so on, so all of them are the same size — which is the first fact about infinity that offends the intuition.

The second fact is that this is not the end. Cantor had shown in 1874, and again with the diagonal argument in 1891, that the real numbers cannot be listed: any proposed list of them leaves at least one out. So there is an infinity strictly larger than ℵ₀. And the argument repeats forever, because the set of all subsets of any set is strictly larger than the set itself. The sizes run ℵ₀, ℵ₁, ℵ₂ and past any indexing that can be written down. There is no largest. There is only a smallest.

There is no largest. There is only a smallest.

That smallest one is the number on the shirt, and the line under it is not a joke. ℵ₀ is the floor of the infinite — the size of the most modest infinite collection that can exist — and every other infinity sits above it. Whether ℵ₁, the next size up, is the size of the real numbers is the continuum hypothesis, and Gödel and Cohen between them proved that the standard axioms of set theory can neither prove it nor refute it. The smallest infinity is pinned down exactly. The one immediately above it is not.