Ordinal Addition by Zero

Theorem
Let $x$ be an ordinal.

Let $\O$ be the zero ordinal.

Then:


 * $x + \O = x = \O + x$

where $+$ denotes ordinal addition.

Proof
By definition of ordinal addition, it is immediate that:


 * $x + \O = x$

We shall use Transfinite Induction on $x$ to prove $\O + x = x$

Base Case
The induction basis $x = \O$ comes down to:


 * $\O + \O = \O$

This follows by the above.

Inductive Case
For the induction step, suppose that $\O + x = x$.

Then, also:

Limit Case
Finally, the limit case.

So let $x$ be a limit ordinal, and suppose that:


 * $\forall y \in x: \O + y = y$

Now we have:

Hence the result, by Transfinite Induction.