\begin{definition}[Definition:Natural Numbers/Von Neumann Construction]
Let $\omega$ denote the minimal infinite successor set.
The '''natural numbers''' can be defined as the elements of $\omega$.
Following Definition 2 of $\omega$, this amounts to defining the '''natural numbers''' as the finite ordinals.
In terms of the empty set $\O$ and successor sets, we thus define:
{{begin-eqn}}
{{eqn | l = 0
      | o = :=
      | r = \O = \set {}
      | c = 
}}
{{eqn | l = 1
      | o = :=
      | r = 0^+ = 0 \cup \set 0 = \set 0
      | c = 
}}
{{eqn | l = 2
      | o = :=
      | r = 1^+ = 1 \cup \set 1 = \set {0, 1}
      | c = 
}}
{{eqn | l = 3
      | o = :=
      | r = 2^+ = 2 \cup \set 2 = \set {0, 1, 2}
      | c = 
}}
{{eqn | o = \vdots
      | c = 
}}
{{eqn | l = n + 1
      | o = :=
      | r = n^+ = n \cup \set n
      | c = 
}}
{{end-eqn}}
This can be expressed in detail as:
{{begin-eqn}}
{{eqn | l = 0
      | o = :=
      | r = \O = \set {}
      | c = 
}}
{{eqn | l = 1
      | o = :=
      | r = \set \O
      | c = 
}}
{{eqn | l = 2
      | o = :=
      | r = \set {\O, \set \O}
      | c = 
}}
{{eqn | l = 3
      | o = :=
      | r = \set {\O, \set \O, \set {\O, \set \O} }
      | c = 
}}
{{eqn | o = \vdots
      | c = 
}}
{{end-eqn}}
\end{definition}