Ordinal Addition by Zero

Theorem
Let $x$ be an ordinal.

Let $\varnothing$ be the zero ordinal.

Then:


 * $x + \varnothing = x = \varnothing + x$

where $+$ denotes ordinal addition.

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


 * $x + \varnothing = x$

We shall use Transfinite Induction on $x$ to prove $\left({\varnothing + x}\right) = x$

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


 * $\varnothing + \varnothing = \varnothing$

This follows by the above.

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

Then, also:

Limit Case
Finally, the limit case.

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


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

Now we have:

Hence the result, by Transfinite Induction.