I know this is a newbie question, so please bare with me :)
I'd like to prove within ZF axiomatic set theory that the addition of two ordinals is not commutative. In particular, I'd like to prove this counter example:
$\omega + 1 \neq 1 + \omega$
I have the following definition for addition on ordinal numbers (defined from transfinite induction):
(i) $\alpha + 0 = \alpha$
(ii) $\alpha + \beta' = (\alpha + \beta)'$
(iii) if $\beta$ is a limit ordinal then $\alpha + \beta = \bigcup_{\gamma \in \beta}(\alpha + \gamma)$
So my attempt was to start from the right side, which, intuitively would be something like this:
$1 + \omega = \bigcup_{\gamma \in \omega}(1 + \gamma) = \{2, 3, 4, ..\}$
My attempt at the left side started like this:
$\omega + 1 = (\omega + 0)' = \omega'$
And then I'm stuck. I'd like to think that the successor of $\omega$ is $\omega$ but with this definition how can I prove that? Also, if that's the case then there's a $1-1$ function that can map $\{1, 2, 3, ...\}$ to $\{2, 3, 4, ...\}$ and still preserve order, so shouldn't both sides of the addition be the same?