2
$\begingroup$

How to construct an ordinal with uncountable cofinality? All the very "large" ordinals I can think of, such as $\omega_\omega^{\omega_\omega}$, still seem to have countable cofinality. I need a better intuitive sense of what such a large ordinal can be.

Relevant links: http://en.wikipedia.org/wiki/Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets http://en.wikipedia.org/wiki/Ordinal_number

  • 0
    @StevenStadnicki yes, you're right, just realized. thanks2012-09-26

1 Answers 1

6

For every ordinal $\alpha$ consider $\alpha+\omega_1$ (ordinal addition). Note that if $\alpha$ is countable (or finite) then the sum is equal to $\omega_1$ which has uncountable cofinality by the virtue of being a regular cardinal. In fact this trick works with any regular uncountable cardinal.

You can always use uncountable cardinals (with uncountable cofinality) as indices, e.g. $\omega_{\omega_{\omega_1}}$

By the way, if you feel that you can construct $\omega_\omega$ which is pretty uncountable, you already have many ordinals with uncountable cofinalities below it.


One note on constructive-ness of ordinals with uncountable cofinality, it requires some choice to prove there exists an ordinal with an uncountable cofinality, since it is consistent with ZF that there are none.

  • 0
    @AsafKaragila, cool, just wanted to make sure I had got it right. thanks again2012-09-27