We can define $0$ to be the number of elements of an empty set.
Then we can define successor of $0$ as the number of all empty sets and we can denote it as our familiar $1$, since there is only one empty set.
Now, we can define successor of $1$, $s(1)$, as the number of $1$-element subsets of $\{0,1\}$ and denote it by our familiar $2$.
More generally, we can define successor of $n$, $s(n)$, as the number of $n$-element subsets of the set $\{0,1,...,n\}$ and this is our $n+1=s(n)$, because from combinatorics we know that ${n+1 \choose n}= {n+1 \choose 1}=n+1$.
What are some other ways to construct naturals, beside those of Conway, Frege/Russel, and von Neumann, and this mine, if this one is legal enough.