Arithmetic of Addition on the Natural Numbers

Let $\mathbb{N}$ be the set of natural numbers, with addition $+$ and successor map $S$ as in that definition. Then the following hold for all $a,b,c\in\mathbb{N}$.

1. $a+1=S(a)$ and $1+a=S(a)$.
2. $S(a)+b=S(a+b)$.
3. (Associativity) $(a+b)+c=a+(b+c)$.
4. (Commutativity) $a+b=b+a$.
5. (Cancellation) If $a+c=b+c$, then $a=b$.
6. If $k\in\mathbb{N}$ and $k\ne 1$, then $k=S(j)$ for exactly one $j\in\mathbb{N}$.
7. $a+b\ne 1$.
8. $a+b\ne a$.