**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 ordinal^{2} 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:

| | | | | Contributors: *bookofproofs* | References: [656]

## 1.**Proposition**: Inequality of Natural Numbers and Their Successors

## 2.**Proposition**: Uniqueness Of Predecessors Of Natural Numbers

## 3.**Proposition**: Addition Of Natural Numbers

## 4.**Definition**: Multiplication of Natural Numbers

[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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