Sup of two ordinals in a limit

Question

I am not sure if this is a dumb question, but if I have a limit ordinal $\xi$, given any two $\alpha,\beta <\xi$, is it possible to find an ordinal $\gamma$ such that $\alpha,\beta\leq\gamma<\xi$?

Thanks!

Answer 1

Ordinals are totally ordered by set membership (at least if we are working with von Neumann ordinals). Consider set union of $\alpha$ and $\beta$. Then,

• If $\alpha \in \beta$, $\alpha \cup \beta = \beta < \xi$.
• If $\beta \in \alpha$, $\alpha \cup \beta = \alpha < \xi$.
• If $\alpha = \beta$, $\alpha \cup \beta = \alpha < \xi$.

So, $\alpha,\beta \leq \alpha \cup \beta < \xi$. $\alpha \cup \beta$ works as $\gamma$ in your question.

Answer 2

Another way to think about it: consider $\gamma=\max(\alpha,\beta)<\xi$. This ordinal satisfies, by definition, $\gamma>\alpha,\beta$ and so it works.

(Note that your property holds true for any set with an order; the fact that $\xi$ is a limit ordinal was not used).