Past exam of the logic and set theory course of the University of Oxford / 2007 / B / 5 / v / Solution

Fix $m$ and induct on $n$. The initial equality $m+0=0+m$ follows from the defining zero clause and (ii). Assuming $m+n=n+m$, the successor clause and (iii) giveThis provesThe extension to arbitrary ordinals is false. By the limit clause for ordinal addition,whereas $\omega +1$ is the successor ordinal of $\omega$ and is strictly greater than it. ThereforeFinite addition is commutative; ordinal addition records the order of concatenated well-orders and need not be commutative.