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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?

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    Show that $\omega+1$ has an element without an immediate predecessor, while $1+\omega$ doesn't.2012-11-19
  • 1
    Or just a maximum.2012-11-19
  • 0
    Yeah, that too :-)2012-11-19

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