proof of commutativity of multiplication for natural numbers using Peano's axiom

Question

How do you prove commutativity of multiplication using peano's axioms.I know we have to use induction and I have already proved n*1=1*n.But I cant think of how to prove the inductive step.

Answer 1

Show that if an operator $\star$ satisfies the defining equations for multiplication, i.e. $0 \star n = 0$ and $(m + 1)\star n = n + m \star n$, then $\star$ is multiplication (this is a straightforward induction).

Then show that the operator defined by $m \star n = n \times m$ satisfies the defining equations for multiplication, and therefore $m \star n = m \times n$, so $n \times m = m \times n$.

That is, show that $n \times 0 = 0$ and $n \times (m + 1) = n + n \times m$, and you're done.

Unfortunately, that's just not easy. $n \times 0 = 0$ is not too bad, you can use induction on $n$, but the only proof I have of the latter statement uses the associativity and commutativity of addition, which both have to be proven by induction as well.