At the beginning of the twentieth century mathematicians wanted to build the whole of mathematics from a single idea, that of a set. But what, in that language, is the number $3$? Georg Cantor had spoken of “ordinals”, numbers that measure positions in a queue: first, second, third, and after all the finite ones, a first infinite one, $\omega$. What was missing was to say what these numbers are, as sets.

In 1923, at 19, von Neumann published a short paper, On the introduction of transfinite numbers, with an answer so economical that it became the standard definition.

Each number is the set of the ones before

There is just one rule: an ordinal is the set of all ordinals smaller than it. Starting from nothing:

$$0 = \varnothing,\qquad 1 = \{0\} = \{\varnothing\},\qquad 2 = \{0, 1\} = \{\varnothing, \{\varnothing\}\},$$$$3 = \{0, 1, 2\} = \{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}.$$

In general, the successor of a number is the number itself with itself added:

$$n + 1 = n \cup \{n\}.$$

Three things come for free with this definition:

  • The number $n$ has exactly $n$ elements: $3$ has the elements $0$, $1$ and $2$. Counting to $n$ means going through the elements of $n$.
  • “Less than” becomes “belongs to”: $m < n$ means $m \in n$. And also “is contained in”: $2 \subset 3$.
  • Everything is built from the empty set. There is no need to assume that numbers exist: they are manufactured out of braces.

Beyond infinity

The same rule carries on. The set of all natural numbers,

$$\omega = \{0, 1, 2, 3, \ldots\},$$

is the first infinite ordinal: it is the set of all ordinals smaller than it, exactly like the earlier ones. Then comes $\omega + 1 = \omega \cup \{\omega\}$, then $\omega + 2$, and so on, up to $\omega \cdot 2$, $\omega^2$, $\omega^\omega$… Cantor had described this staircase; von Neumann gave each step a concrete construction.

Two years later, in his doctoral thesis, von Neumann proposed a system of axioms for set theory that distinguished sets from classes, collections too large to be sets, such as “all sets”. With adjustments by Paul Bernays and Kurt Gödel, this system became known as NBG, and it is still one of the two most widely used ways of founding mathematics.

Exercise: write 4

Write the ordinal $4$ using only the earlier numbers, then count the opening braces $\{$ that appear when everything is written with $\varnothing$.

$4 = \{0, 1, 2, 3\}$. Written only with $\varnothing$, each number $n$ uses one outer brace plus the braces of its elements. Calling $c(n)$ the number of opening braces, $c(0) = 0$ and $c(n+1) = 1 + c(0) + c(1) + \cdots + c(n)$. So $c(1) = 1$, $c(2) = 2$, $c(3) = 4$, $c(4) = 8$: the number of braces doubles at each step, $c(n) = 2^{n-1}$.

Exercise: less than is belonging

Check that $1 \in 3$ and $1 \subset 3$, but $3 \notin 1$.

$3 = \{0, 1, 2\}$ contains $1$ as an element, so $1 \in 3$. And $1 = \{0\}$, whose only element, $0$, is also in $3$, so $1 \subset 3$. But $1 = \{0\}$ does not have $3$ among its elements, so $3 \notin 1$, as it should be, since $3$ is not less than $1$.

Sources: John von Neumann, “Zur Einführung der transfiniten Zahlen”, Acta Sci. Math. Szeged (1923) · Jean van Heijenoort (ed.), From Frege to Gödel (1967) · George Pólya, The Pólya Picture Album (1987).

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