Proof That Addition of Ordinal Numbers is Associative

Suppose that A, B and C are three muturally disjoint sets and suppose that

Thenand

where well orderingmeans 'A is less than B' and well orderingmeans 'B is less than C'. Applying these successively we obtain

and

butandboth mean exactly the same, namely that A is less than B and B is less than C, so we can write hence

Ifwe can write

Ifwe can write

Ifwe can write