## 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:

$x + \varnothing = x$

$\Box$

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.

$\Box$

### Inductive Case

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

Then, also:

 $\displaystyle x^+$ $=$ $\displaystyle \left({\varnothing + x}\right)^+$ Substitutivity of Equality $\displaystyle$ $=$ $\displaystyle \varnothing + x^+$ Definition of ordinal addition

$\Box$

### 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:

 $\displaystyle x$ $=$ $\displaystyle \bigcup_{y \mathop \in x} y$ Union of Limit Ordinal $\displaystyle$ $=$ $\displaystyle \bigcup_{y \mathop \in x} \left({\varnothing + y}\right)$ Indexed Union Equality $\displaystyle$ $=$ $\displaystyle \varnothing + x$ Definition of ordinal addition

$\Box$

Hence the result, by Transfinite Induction.

$\blacksquare$