Natural Numbers
Why
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?
Definition
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.
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:
The set of natural numbers is the minimal successor set. A natural number (or number, natural) is an element of this minimal successor set.
Notation
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 $n - \set{0} = \upto{n}$.