I know the statement "if $\alpha<\beta$, then $\alpha + \gamma < \beta + \gamma$ " is wrong, since $0<1$ but $0+\omega = \omega = 1 + \omega$. But what about "if $\alpha<\beta$, then $\gamma + \alpha < \gamma + \beta$ " ?
Here is definitions I used (Thomas Jech, Set Theory):
$\alpha + 0 = \alpha$ for all $\alpha\in\mathrm{Ord}$,
$\alpha+(\beta + 1) = (\alpha + \beta) + 1$ for all $\alpha,\beta\in\mathrm{Ord}$,
$\alpha + \beta = \lim_{\xi\to\beta}(\alpha + \xi)$ for $\alpha,\beta\in\mathrm{Ord}$ and $\beta$ is a limit ordinal.
and in general, limit is defined as $\lim_{\xi\to\beta}\gamma_\xi=\sup\{\gamma_\xi: \xi<\beta\}$ only if the sequence $\gamma_\xi$ is nondecreasing and $\beta$ is a limit ordinal.
I think that the statement is true, since I get there when I was trying to prove $\alpha + \gamma = \lim_{\xi\to\gamma}(\alpha+\xi)$ is defined well when $\gamma$ is a limit ordinal. Firstly, I tried to prove the sequence $\langle\alpha + \xi: \xi<\gamma\rangle$ is increasing since the limit is defined only if the sequence is nondecreasing, in Thomas Jech's book. But I have to show that "if $\xi_1<\xi_2$, then $\alpha+\xi_1<\alpha+\xi_2$" to show that the sequence is increasing (of course showing that $\alpha+\xi_1\leq\alpha+\xi_2$ is enough but I believe that $\alpha+\xi_1<\alpha+\xi_2$).
I tried to prove the statement, and I'm stuck.
Let $\Gamma$ be the class of all ordinals $\gamma$ satisfying the statement "$\forall\alpha,\beta\in\mathrm{Ord}\left(\alpha<\beta \Longrightarrow\gamma + \alpha < \gamma + \beta\right)$ ".
$(i)$ $0\in\Gamma$, since $0 + \eta = \eta$ for all $\eta\in \mathrm{Ord}$. (I previously proved this).
$(ii)$ Assume that $\gamma\in\Gamma$. Then $(\gamma + 1) + \alpha = \gamma + (1 + \alpha) <^? \gamma + (1+\beta) = (\gamma + 1) + \beta$ (But I couldn't show the inequality).
$(iii)$ Assume that for all ordinals $\xi<\gamma$, $\xi\in\Gamma$, i.e., if $\alpha<\beta$ then $\xi+\alpha < \xi+\beta$. Then, $\gamma+\alpha$… (and that is it, i couldn't continue, since definition doesn't say anything about addition when limit ordinal is at the left side)
Thanks a lot!
Induction on $\beta$ worked well!
And I love the limit case of the transfinite induction:
Let $\beta$ be a limit ordinal and "if $\alpha<\xi$ then $\gamma + \alpha < \gamma + \xi$" for all $\xi<\beta$.
This means the sequence $\langle\gamma + \xi : \xi<\beta\rangle$ is increasing, so we can use the definition of limit: $\gamma + \beta = \lim_{\xi\to\beta}(\gamma + \xi) = \sup\{\gamma+\xi:\xi<\beta\}$
Also, we know that there are ordinals $\theta_1$ and $\theta_2$ such that $\alpha<\theta_1<\theta_2<\beta$, if $\alpha < \beta$, since $\beta$ is a limit ordinal. Notice that $\gamma + \theta_1,\gamma + \theta_2\in\{\gamma + \xi: \xi < \beta\}$ so $\gamma + \theta_1<\gamma + \theta_2\leq\sup\{\gamma + \xi: \xi<\beta\}$.
so, if $\alpha<\beta$, then $\gamma+\alpha<\gamma+\theta_1<\sup\{\gamma + \xi: \xi<\beta\} = \lim_{\xi\to\beta}(\gamma + \xi) = \gamma +\beta$.
Best Answer
This is very easy to understand when using the order-theoretic definition of $\alpha+\beta$ as the linear order which is the initial segment $\alpha$, followed by the tail segment $\beta$.
Remember now that there is at most a single embedding of one ordinal into another whose image is an initial segment.
We note that $\gamma$ is a joint initial segment of both $\gamma+\alpha$ and $\gamma+\beta$, so the identity function is the only embedding. Now we may proceed by considering the embedding of $\alpha$ into $\beta$, which itself is also the identity, and using it to extend the embedding of $\gamma\to\gamma$.
This shows, easily, that $\gamma+\alpha\leq\gamma+\beta$. But now we remember that the embedding $\alpha\to\beta$ was not surjective, so the embedding we got is not surjective. Since that is the only embedding onto an initial segment, it must be that $\gamma+\alpha<\gamma+\beta$.
Of course, this doesn't help you if you're trying to prove the inequality from the recursive definition. But it gives a good picture as to what's going on.
Now fix $\gamma$, and prove by induction on $\beta$, that for all $\alpha<\beta$, $\gamma+\alpha<\gamma+\beta$; and conclude this is true for all $\alpha,\beta,$ and $\gamma$.