Braces Of Eight (Artwork)
Braces Of Eight Digital Print 2021

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.

Braces of Eight, detail.

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