Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

Thank you.

share|cite|improve this question
Take $\beta=\alpha+\omega$, where the sum is ordinal addition. – Brian M. Scott Jan 9 '13 at 1:35
up vote 2 down vote accepted

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.