The number eight shown as a set-theoretic finite ordinal number using the construction of the non-negative integers described by John von Neumann around 1923.

### Background and Inspiration

How does one define the concept of number? What is the meaning of a number such as eight? One way to do this is from the perspective of a set.

In this set-theoretic finite ordinal number representation, zero is represented by the empty set: \(\{\}\) or \(\emptyset\). Given \(0 = \{\emptyset\}\), we can define the integers recursively with the use of a successor function, \(s(n)\), defined as follows: \[s(n) = n + 1 = n \cup \{n\}.\] One, the successor of 0, is the set containing zero (the empty set): \(\{\{\}\}\) or \(\{\emptyset\}\). Two, the successor of 1, is the set containing zero and one: \(\{ \{\}, \{\{\}\}\}\) or \(\{\emptyset, \{\emptyset\}\}\). In general, the number \(n\) is the successor of \(n-1\) and the set containing the numbers \(0\) through \(n-1\): \(\{0, 1, 2, \ldots, n-1\}\). In this piece, the empty set \(\emptyset\) is depicted as a black square and sets are depicted by rectangles.

An interesting property of this construction is that \(a < b\) can be defined using set operations. Let \(a\) and \(b\) be two numbers and \(A\) and \(B\) be the two sets representing them, respectively, then \(a < b\) if and only if \(A \subset B\).

You can see this pattern by examining the the first five integers expressed in set notation: \[\begin{aligned} 0 &= \emptyset\\ 1 &= \{\emptyset\}\\ &= \{0\}\\ 2 &= \{\emptyset,\ \{\emptyset\}\}\\ &= \{0, 1\}\\ 3 &= \{\emptyset,\ \{\emptyset\},\ \{\emptyset,\{\emptyset\}\}\}\\ &= \{0, 1, 2\}\\ 4 &= \{\emptyset,\ \{\emptyset\},\ \{\emptyset,\{\emptyset\}\},\ \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}\}\\ &= \{0, 1, 2, 3\}\\ 5 &= \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}\}\}\\ &= \{0, 1, 2, 3, 4\}\\ \end{aligned}\]

Furthermore, the number of elements of each set (its cardinality) is just the number itself. For example, the set representing 8 has 8 elements.