Is there an unbounded countable set of ordinals?

Question

Let $\text{Ord}=\{0,1,\ldots,\omega,\ldots\}$ be the set of ordinals. Does $\text{Ord}$ have an unbounded countable subset? In particular, is $$\{\omega, \omega^\omega, \omega^{\omega^\omega},...\}$$ unbounded?

Answer 1

Your construction itself almost constitutes a proof that the supremum of that set exists by transfinite induction.

• $F(0) = \omega$
• $F(a+1) = \omega^{F(a)}$
• $F(b) = \bigcup_{x<b} F(x)$ when $b$ is a limit ordinal.

The supremum of your set is $F(\omega)$