We want to show something holds for every
natural number.^{1}

The most important property of the set of natural numbers is that it is the unique smallest successor set. In other words, if $S$ is a successor set contained in $\omega $ (see Natural Numbers), then $S = \omega $. This is useful for proving that a particular property holds for the set of natural numbers.

To do so we follow standard routine. First, we define the set $S$ to be the set of natural numbers for which the property holds. This step uses the principle of selection (see Set Selection) and ensures that $S \subset \omega $. Next we show that this set $S$ is indeed a successor set. The first part of this step is to show that $0 \in S$. The second part is to show that $n \in S \implies \ssuc{n} \in S$. These two together mean that $S$ is a successor set, and since $S \subset \omega $ by definition, that $S = \omega $. In other words, the set of natural numbers for which the property holds is the entire set of natural numbers. We call this the principle of mathematical induction.

- Future editions will modify this superficial why. ↩︎