# Show that a particular set is not a limit ordinal (aim is to define ordinal subtraction)

Let $\alpha$ and $\beta$ be two ordinals with $\beta \leq \alpha$. Define

$$X:= \{\gamma \in \alpha^{+} : \beta + \gamma \leq \alpha\}.$$

I have shown this is an ordinal. Now I need to show it isn't a limit ordinal, but I'm stuck.

• What is $\alpha^+$ for you? (Because in many contexts, that denotes the successor cardinal of $|\alpha|$.) – Asaf Karagila May 19 '15 at 20:37
• It is the successor of $\alpha$, namely $\alpha \cup \{ \alpha\}$. – Frank May 19 '15 at 20:37
HINT: Assume to the contrary, and use the definition of $\beta+\delta$ when $\delta$ is a limit ordinal, to show that $\beta+\delta\leq\alpha$ as well to obtain a contradiction.
• Ok thanks. If this is what the writer of the exercise intended I would be surprised, as the next part uses this $\gamma$ as 'another' way of doing the problem. – Frank May 19 '15 at 20:41