Set-theoretic Definitions of Natural Numbers Ernst Zermelo 1908, John von Neumann 1923

Definition: Set-theoretic Definitions of Natural Numbers (Ernst Zermelo 1908, John von Neumann 1923)

The set of natural numbers \(\mathbb N\) is defined using the concept ordinals, as follows:

Definition due to von Neumann (1923)

(1) The empty set (as the first ordinal)¹ represents the first natural number:

\[0:=\{\emptyset\}.\]

(2) Once we have the ordinal \(n=\alpha\), we can construct a bigger ordinal² using recursively the formula for constructing successors of ordinals, denoting the successor \(n^+\) of the natural number \(n\):
\[n^+:=s(\alpha):=\alpha\cup\{\alpha\}=n\cup \{n\}.\]
Applying the set axioms and this construction systematically, it gives us a chain of ordered ordinals

\[\begin{array}{rcl}0&:=&\emptyset,\\1&:=&0\cup\{0\}=\emptyset\cup\{\emptyset\}=\{\emptyset\},\\2&:=&1\cup\{1\}=\{\emptyset\}\cup\{\{\emptyset\}\}=\{\emptyset,\{\emptyset\}\},\\3&:=&2\cup\{2\}=\{\emptyset,\{\emptyset\}\}\cup\{\{\emptyset,\{\emptyset\}\}\}=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\},\\&\vdots&\\n^+&:=&n\cup\{n\},\\&\vdots&\end{array}\]

which can be visualized in the following figure

and for which we introduce the notation \(0,1,2,3,\ldots\):

\[0 < 1 < 2 < 3 < \ldots\]

Due to the axiom of infinity we can postulate the existence of an infinite set, which is “contains” all such sets.³

\[\mathbb N:=\bigcup n=\{0,1,2,3,\ldots.\}\]

¹ Please note that it is well defined due to the axiom of existence of empty set.

² Ordinals are sets with some interesting properties, including trichotomy, ensuring that all ordinals can be compared with each other by the relation
\[\alpha < \beta:\Leftrightarrow \alpha\in\beta.\]
For any two ordinals, and in particular for natural numbers, we can therefore always decide which one is “bigger”, “smaller”, or whether they are equal to each other.

³ Please note that this infinite set is an ordinal by definition. However, we have not built by the above construction formula, i.e. it is not a successor of any “previous” ordinal. In other words, \(\mathbb N\) is the first limit ordinal.

Definition due to Ernst Zermelo (1908)

The set \(\mathbb N\) of natural numbers is defined recursively by: \[\begin{array}{rcl}0&:=&\emptyset,\\1&:=&\{0\}=\{\emptyset\},\\2&:=&\{1\}=\{\{\emptyset\}\},\\3&:=&\{2\}=\{\{\{\emptyset\}\}\},\\&\vdots&\\n^+&:=&\{n\}=\underbrace{\{\ldots\{ }_{n+1\text{ times}}\emptyset\underbrace{\}\ldots\} }_{n+1\text{ times}},\\&\vdots&\\\end{array}\]

This definition can be visualized as follows: