Successor Sets
Why
We want numbers to count with.1
Definition
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.
Notation
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
\[
\begin{aligned}
0 &= \varnothing \\
1 &= \ssuc{0} = \set{0} \\
2 &= \ssuc{1} = \set{0, 1} \\
3 &= \ssuc{2} = \set{0, 1, 2} \\
4 &= \ssuc{3} = \set{0, 1, 2, 3} \\
\end{aligned}
\]