Proving of existence of limit ordinal

Question

How to prove that for every ordinal $\alpha$ there exist limit ordinal $\beta$ ,such that$\alpha\in\beta$ ?

Thank you.

Answer 1

You have several options.

1. If you already proved basic things about ordinal addition, $\alpha+\omega$.
2. If you have proved basic things about cardinals, $\alpha^+$.
3. If you have to write full proof, show that there is a definable function from $\omega$ which sends $n$ to the $n$-th successor of $\alpha$. Use replacement and union to prove the existence of $\alpha+\omega$.

Also related: LIM is cofinal in ON