Ordinal inequalities I have a problem with the following exercise:
First a definition:
$ \mathord{\in} = \{\langle x,y \rangle; x \in y \}$
Exercise: Show that the following conditions are equivalent:
a. For every $\beta,\gamma < \alpha$, $\beta + \gamma < \alpha$
b. For every $\beta < \alpha$, $\beta + \alpha = \alpha$
c. For every $A \subseteq \alpha$, $\operatorname{order-type}(\langle A,\in\rangle) = \alpha$, or, $\operatorname{order-type}(\langle A \setminus \alpha,\in \rangle) = \alpha$ 
d. There exists an ordinal $\delta$ such that, $\alpha = \omega^\delta$
Anyone have a hint for me? Especially for c. I tried to prove $a \implies b$ by induction, but I am not sure whether the induction should be on $\alpha$ or on $\gamma$.
I also don't quite get the intuition behind this. Especially, the intuition of part c.
Thank you!!
Shir 
 A: First, a minor point, you should assume $\alpha\gt0$; if $\alpha=0$, then the first three statements are true, but the last is false.
(a)$\Rightarrow$(b): Suppose $\beta\lt\alpha$. Clearly $\alpha\le\beta+\alpha$; assume for a contradiction that $\alpha\lt\beta+\alpha$. Then we have $\beta\lt\alpha\lt\beta+\alpha$, which implies that $\alpha=\beta+\gamma$ for some ordinal $\gamma\lt\alpha$. Now we have $\beta,\gamma\lt\alpha$ and $\beta+\gamma=\alpha$, contradicting (a).
(b)$\Rightarrow$(a): $\beta+\gamma\lt\beta+\alpha=\alpha$.
(c)$\Rightarrow$(b): Suppose $\beta\lt\alpha$. Write $\alpha=\beta+\gamma$. Applying (c) with $A=\beta$, either $\beta=\alpha$ or $\gamma=\alpha$. Since $\beta\lt\alpha$, we must have $\gamma=\alpha$ and $\beta+\alpha=\beta+\gamma=\alpha$.
(a)$\Rightarrow$(d): We have $\omega^{\delta}\le\alpha\lt\omega^{\delta+1}$ for some ordinal $\delta$. Assume for a contradiction that $\omega^{\delta}\lt\alpha\lt\omega^{\delta+1}$. By (a) we have $\omega^{\delta}2\lt\alpha$ and, by induction, $\omega^{\delta}n\lt\alpha$ for all $n\lt\omega$, whence $\alpha\ge\omega^{\delta}\omega=\omega^{\delta+1}$, a contradiction.
(d)$\Rightarrow$(c): Proof by induction on $\delta$. The straightforward details are left to the reader. By the way, what we are proving here is the partition relation $\omega^{\delta}\rightarrow(\omega^{\delta},\omega^{\delta})^1$ in the infamous "arrow notation" of Erdős and Rado.
