2
$\begingroup$

I've got a most probably silly question, but I can't find the answer to it:

If $\alpha$ is a countable limit ordinal, how can we be sure that there exist ordinals $\alpha_n$ such that $\alpha = \bigcup \{\alpha_n\mid n\in\omega\}$ (rather than the usual $\alpha = \bigcup \{\beta\mid \beta<\alpha\}$)?

Cheers!

$\endgroup$

1 Answer 1

3
$\begingroup$

Well, first of all $\{\beta\mid\beta<\alpha\}=\alpha$ is countable, therefore it can be enumerated as $\alpha_n$'s. But I suppose that you want to ask about a strictly increasing sequence.

First enumerate $\alpha$ as $\{\beta_n\mid n<\omega\}$. Then by induction define $\alpha_n$, where $\alpha_{n+1}=\beta_k$ where $k=\min\{m\mid\alpha_n<\beta_m\}$. You can now show that this is a strictly increasing and cofinal sequence.

$\endgroup$
6
  • $\begingroup$ Thanks for the answer. I'm new to ordinals and while I know (knew) that countable things can be enumerated as $\alpha_n$'s, I am now unsure why we can do so using only $\omega$, and not, say, $\omega^2$ indices to do so. Is that the definition of countability? Sorry for all the stupid questions. $\endgroup$
    – Fedra
    Commented May 5, 2014 at 19:19
  • $\begingroup$ Well, being countable is by definition to have an injection into $\omega$, or in other words to be enumerated. $\endgroup$
    – Asaf Karagila
    Commented May 5, 2014 at 19:21
  • 1
    $\begingroup$ Okay, in this case I'm fine. It's just weird how I am now unsure about so many things in set theory I didn't even think I could be unsure about a few weeks ago... :-) $\endgroup$
    – Fedra
    Commented May 5, 2014 at 19:23
  • 2
    $\begingroup$ It's the graph of learning. Wait, you'll be sure again and then unsure again and then sure again... and so on and so forth. $\endgroup$
    – Asaf Karagila
    Commented May 5, 2014 at 19:27
  • 1
    $\begingroup$ You're welcome. I'm probably gonna help you again in the future. $\endgroup$
    – Asaf Karagila
    Commented May 5, 2014 at 19:39

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .