Question: If $\Theta$ is the order type of an uncountable set, then show that for $\alpha < \omega_1$, $\alpha \preceq \Theta$ or $\alpha^* \preceq \Theta.$ Where $\preceq$ is an ordering of order types.
This makes sense to me conceptually. For every order type less than the first uncountable order type, this order type must be less than any uncountable order type. Is this way of thinking correct?