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

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


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

Notation: $\displaystyle \omega_1$ is the first uncountable ordinal. $\displaystyle \text{seg}$ 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.