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

Let $\alpha$ an ordinal and $\langle\alpha_\xi\rangle$ a cofinal sequence of elements of $\alpha$. The length, $\gamma$, of this sequence is at least $\operatorname{cf}\alpha$ but can be equal to any ordinal $\geq \operatorname{cf}\alpha$. If this sequence is a strictly increasing one then $\gamma$ is $\leq$ to the order type of $\alpha$ that is $\alpha$. Is it exact ?

share|cite|improve this question
up vote 3 down vote accepted

If $\alpha$ is a limit ordinal and $\langle\alpha_\xi:\xi<\gamma\rangle$ is a non-decreasing sequence cofinal in $\alpha$, then $\operatorname{cf}\gamma=\operatorname{cf}\alpha$, even if the sequence is not strictly increasing. As you say, if the sequence is strictly increasing, then $\gamma\le\alpha$.

share|cite|improve this answer
Thanks. I forgot Jech, set theory p.32 ... – Marc Moretti Nov 18 '12 at 21:49
@Marc: You’re welcome. – Brian M. Scott Nov 18 '12 at 21:50

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.