What are numbers? We want to count, forever. Does a set exist which contains zero, and one, and two, and three, and all the rest?

In Successor Sets, we said “and we continue as usual using the English language...” in our definition of zero, and one and two and three. Can this really be carried on and on? We will say yes. We will say that there exists a set which contains zero and contains the successor of each of its elements.

A set which contains 0 and contains the
successor of each of its elements exists.

This principle is sometimes called the
principle of infinity (or
axiom of infinity).
We want this set to be unique.
The principle says one successor set exists,
but not that it is unique.
To see that it is unique, notice that the
intersection of a nonempty family of successor
sets is a successor set.^{1}
Consider the intersection of the family of all
successor sets.
The intersection is nonempty by the principle
of infinity (see Intersection of Empty Set for this subtlety).
The principle of extension guarantees that this
intersection, which is a successor set contained
in every other successor set, is unique.
We summarize:

There exists a unique smallest successor set.

The set of natural numbers is the minimal successor set. A natural number (or number, natural) is an element of this minimal successor set.

We denote the unique smallest successor set by
$\omega $.^{2}
We denote the set of natural numbers without 0
by $\N $, a mnemonic for natural.
In other words $\N = \omega - \set{0}$.
We often denote elements of $\omega $ or $\N $
by $n$, a mnemonic for number, or $m$, the
letter before $m$ in the conventional ordering
of the Latin alphabet (see Letters).

We denote the natural numbers up to $n$ by
$\upto{n}$.
Recall that $n$ *is a set*.
In other words, we have defined $n$ so that
$n - \set{0} = \upto{n}$.