Successor ordinal is preserved under addition

Question

Is it true that if $a$ is an ordinal and $b$ a successor ordinal, then $a+b$ is a successor ordinal? If so, how does the proof go? It seems true to me, but I can't prove it.

Answer 1

Every well-order is isomorphic to a unique ordinal.

For ordinals $a,b$ the ordinal $a+b$ is defined as the ordinal that is isomorphic to the lexicographic (dictionary) order on $S=(\{0\}\times a)\cup (\{1\}\times b).$ If $b=c+1$ then $(1,c)$ is the largest member of $S$ , so $a+b$ has a largest member, so $a+b$ is a successor ordinal.

Intuitively, $a+b$ is formed by putting a copy of $b$ "after" $a$.