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: \(\{\}\). Starting with \(0 = \{\}\), we can define the integers recursively with the use of a successor function, \(s(n)\), defined as follows: \[s(n) = n + 1 = n \cup \{n\}.\] The successor of 0, \(s(0)\), is 1: \[1 = s(0) = 0 \cup \{0\} = \{\} \cup \{\{\}\} = \{\{\}\}.\] The successor of 1, \(s(1)\), is 2: \[2 = s(1) = 1 \cup \{1\} = \{\{\}\} \cup \{\{\{\}\}\} = \{ \{\}, \{\{\}\}\}.\] The successor of 2, \(s(2)\), is 3: \[\begin{aligned} 3 = s(2) = 2 \cup \{2\} &= \{ \{\}, \{\{\}\}\} \cup \{\{ \{\}, \{\{\}\}\}\} \\ &= \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}.\\ \end{aligned}\] The successor of 3, \(s(3)\), is 4: \[\begin{aligned} 4 &= s(3) \\ &= 3 \cup \{3\} \\ &= \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\} \cup \{\{\{\},\{\{\}\}, \{\{\},\{\{\}\}\}\}\} \\ &= \{\{\}, \{\{\}\}, \{\{\},\{\{\}\}\},\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\}.\\ \end{aligned}\] In general, the number \(n\) is the successor of \(n-1\) and is the set containing the numbers \(0\) through \(n-1\): \[\begin{aligned} s(n-1) &= (n-1) \cup \{(n-1)\}\\ &= \underbrace{(n-2) \cup \{(n-2)\}}_{(n-1)} \cup \{(n-1)\}\\ &= \underbrace{(n-3) \cup \{(n-3)\}}_{(n-2)} \cup \{(n-2)\} \cup \{(n-1)\}\\ &= \cdots\\ &= 0 \cup \{(0)\} \cup \{(1)\} \cup \{(2)\} \cup \cdots \cup \{(n-1)\} \\ &= \{\} \cup \{0, 1, 2, \ldots, n-1\}.\\ &= \{0, 1, 2, \ldots, n-1\}.\\ \end{aligned}\]

A nice feature of this representation is that the number of elements of each set (its cardinality) is just the number itself. For example, the set representing 8 is \(\{0,1,2,3,4,5,6,7,\}\), which has 8 elements. Note these elements are generally sets containing other elements that are sets.

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 \(A \subset B\), where is \(A\) is a proper subset of \(B\).

Note the commas in the representation are extraneous, so juxtaposition is sufficient in expressing the ordinals. The number of \(\{\) symbols needed to represent \(n\) is \(2^n\). Similarly, \(2^n\) \(\}\) symbols are needed, resulting in a total of \(2^{n+1}\) required symbols. Thus 512 total symbols are needed to represent 8.

The artwork *Braces of Eight* uses just the symbols \(\{\) and \(\}\) to represent eight.
The thickness of the symbols are varied to show the numeral 8.
The background texture is made from randomly sized, colored, and placed 8's
as shown below.