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)<br />

and

<br />
\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.