4
$\begingroup$

I need a little help regarding one of the basic properties of $\omega _1$.

Namely, I am supposed to show that for any $\alpha \in \omega_1$ there exists $\beta \in \omega_1\cap A$ such that $\alpha < \beta$.

Here $A$ stands for a set $A:=$ { $\alpha$ $ |$ $ \alpha \in Ord$ $\wedge$ $cof(\alpha)=\omega$ }.

What I am trying to show is that for any $\alpha \in \omega_1$ $\exists \beta \in \omega_1 \cap Lim$, where $Lim$ is the class of all limit ordinals such that $\alpha < \beta$.

This will be enough since every countable limit ordinal has cofinality $\omega$ which is not so hard to show.

My problem is, how to show that this defined $\beta$ indeed exists, i.e. $\beta$ being $Lim$.

Improve this question
$\endgroup$
  • 2
    $\begingroup$ How about letting $\beta = \sup \{ \alpha + 1, \alpha + 2, \cdots \}$? $\endgroup$ – Clive Newstead Mar 18 '17 at 12:22
  • $\begingroup$ Okay, it is a limit ordinal, but what is the guarantee for being strictly < than $\omega_1$? $\endgroup$ – edward_scissorhands Mar 18 '17 at 12:26
  • 1
    $\begingroup$ It's a countable union of countable sets, so is countable. $\endgroup$ – Clive Newstead Mar 18 '17 at 12:29
  • $\begingroup$ Thanks! It makes perfect sense now. $\endgroup$ – edward_scissorhands Mar 18 '17 at 12:30
  • $\begingroup$ I'll convert my comments into an answer. $\endgroup$ – Clive Newstead Mar 18 '17 at 12:30
5
$\begingroup$

Define $\beta = \sup \{ \alpha + n \mid n < \omega \}$. Then $\beta > \alpha$ and $\beta$ has cofinality $\omega$; moreover, it is less than $\omega_1$ since it is a countable union of countable sets, so is countable.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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