We want numbers to count with.^{1}

The successor of a set is the set which is the union of the set with the singleton of the set. In other words, the successor of a set $A$ is $A \cup \set{A}$. This definition is primarily of interest for the particular sets introduced here.

These sets are the following (and their
successors):
We call the empty set
zero.^{2}
We call the successor of the empty set
one.
In other words, one is $\varnothing \cup
\set{\varnothing} = \set{\varnothing}$.
We call the successor of one
two.
In other words, two is $\set{\varnothing} \cup
\set{\set{\varnothing}} = \set{\varnothing,
\set{\varnothing}}$.
Likewise, the successor of two we call
three and the successor
of three we call four.
And we continue as usual,^{3}
using the English language in the typical way.

A set is a successor set if it contains zero and if it contains the successor of each of its elements.

Let $x$ be a set. We denote the successor of $x$ by $\ssuc{x}$. We defined it by

\[ \ssuc{x} \coloneqq x \cup \set{x} \]

We denote one by $1$. We denote two by $2$. We denote three by $3$. We denote four by $4$. So

\[ \begin{aligned} 0 &= \varnothing \\ 1 &= \ssuc{0} = \set{0} \\ 2 &= \ssuc{1} = \set{0, 1} \\ 3 &= \ssuc{2} = \set{0, 1, 2} \\ 4 &= \ssuc{3} = \set{0, 1, 2, 3} \\ \end{aligned} \]