Ordinal Addition is Associative

Theorem
Ordinal addition is associative, i.e.:


 * $\left({x + y}\right) + z = x + \left({y + z}\right)$

holds for all ordinals $x$, $y$ and $z$.

Proof
By Transfinite Induction on $z$.

Basis for the Induction
This proves the basis for the induction.

Induction Step
Suppose that:


 * $x + \left({y + z}\right) = \left({x + y}\right) + z$

Then:

This proves the induction step.

Limit Case
Let $z \in K_{II}$ be a limit ordinal, and suppose that:


 * $\forall w \in z: x + \left({y + w}\right) = \left({x + y}\right) + w$

Then it follows that:

By the hypothesis, we have that $z \in K_{II}$.

From Limit Ordinals Preserved Under Ordinal Addition, also $y + z \in K_{II}$.

Therefore, by definition of ordinal addition:


 * $x + \left({y + z}\right) = \displaystyle \bigcup_{n \mathop \in y + z} \left({x + n}\right)$

So it is sufficient to prove that:


 * $\displaystyle \bigcup_{w \mathop \in z} \left({x + \left({y + w}\right) }\right) = \bigcup_{n \mathop \in y + z} \left({x + n}\right)$

Take any $w \in z$.

Then by Membership is Left Compatible with Ordinal Addition, $y + w < y + z$.

Setting $n = y + w$, we now have that:


 * $x + \left({y + w}\right) \le x + n$

By Supremum Inequality for Ordinals, it now follows that:


 * $\displaystyle \bigcup_{w \mathop \in z} \left({x + \left({y + w}\right) }\right) \le \bigcup_{n \mathop \in y + z} \left({x + n}\right)$

To prove the converse inequality, take any $n \in y + z$.

By definition of ordinal addition, this means:


 * $n \in \displaystyle \bigcup_{w \mathop \in z} \left({y + w}\right)$

and so, for some $w \in z$, we have $n \in y + w$.

By Membership is Left Compatible with Ordinal Addition, this yields:


 * $x + n \in x + \left({y + w}\right)$

and whence by Supremum Inequality for Ordinals:


 * $\displaystyle \bigcup_{n \mathop \in y + z} \left({x + n}\right) \le \bigcup_{w \mathop \in z} \left({x + \left({y + w}\right) }\right)$

By definition of set equality:


 * $\displaystyle \bigcup_{n \mathop \in y + z} \left({x + n}\right) = \bigcup_{w \mathop \in z} \left({x + \left({y + w}\right) }\right)$

This proves the limit case.

Hence the result, by Transfinite Induction.

Also see

 * Natural Number Addition is Associative/Proof 2