A limit ordinal $\gamma$ is indecomposable if and only if $\alpha + \gamma = \gamma$ if and only if $\gamma = \omega^{\alpha}$ for some $\alpha$.
136081
Indecomposable limit ordinals
94365
How to prove that if $D$ is countable, then $f(D)$ is either finite or countable?
134844
Summablity and countability
62936
Proving equivalencies of two versions of the Axiom of Choice
78290
Countable axiom of choice: why you can't prove it from just ZF
11912
Is it true that $\displaystyle \left( \lim_{n \to \infty} x_n \right)^\alpha = \lim_{n \to \infty} x_n^\alpha $
66398
Union of a finite set and a countably infinite set is countably infinite
148516
Proof that $\omega^\omega$ is completely metrizable and second countable
78941
proof that an alternative definition of limit is equivalent to the usual one
97583
Let $\alpha, \beta, \gamma$ be cardinals, $\beta \leq \gamma$, prove $\alpha ^{\beta}\le \alpha ^{\gamma}$
137768
Limit Ordinal Question
104190
Can any infinite ordinal be expressed as the sum of a limit ordinal and a finite ordinal?
72920
$x \perp y$ if and only if $\Vert x + \alpha y \Vert \ge \Vert x \Vert$ for all scalars $\alpha$
43219
If $[K(\alpha):K]=p\neq q=[K(\beta):K]$ then $[K(\alpha+\beta):K]=pq$
150882
If $\Gamma \cup \{ \neg \varphi \}$ is inconsistent, then $\Gamma \vdash \varphi$
51134
If $E \in \sigma(\mathcal{C})$ then there exists a countable subset $\mathcal{C}_0 \subseteq \mathcal{C}$ with $E \in \sigma(\mathcal{C}_0)$
90471
Can the limit of a product exist if neither of its factors exist?