## initial segments

Show that there exists a $\gamma \in \omega_1$ such that for all $\alpha < \gamma$ and $\beta < \gamma$,

$\text{seg}_{\omega_1}(\alpha) +_o \text{seg}_{\omega_1}(\beta) <_o \text{seg}_{\omega_1}(\gamma)
$

and

$
\text{seg}_{\omega_1}(\alpha) \cdot_o \text{seg}_{\omega_1}(\beta) <_o \text{seg}_{\omega_1}(\gamma)$
.

Notation: $\omega_1$ is the first uncountable ordinal. $\text{seg}$ denotes the initial segment. We may use the Countable Principle of Choice, but not the Axiom of Choice. I do not see how to do this. I just need a few hints on how to proceed.