Ordinal Arithmetic: Proving X is Countably Compact

[Gear300]

Problem Statement:
Show that the set X of all ordinals less than the first uncountable ordinal is countably compact but not compact.

Let μ be the first uncountable ordinal.

The latter question is easy to show, but I stumbled upon a curiosity while attempting the former. In showing the former, I simply tried to show that every infinite subset of X should have a limit point (or in particular, an ω-accumulation point) in X. And so, in doing this, I needed to ensure that any infinite subset with μ as a limit point has another limit point in X. I reasoned that the first ω ordinals of this subset should only span a countable range of ordinals, since each of their co-initials are countable and a countable union of countable sets is countable. Any neighborhood of μ, however, is uncountable, so the limit point of the first ω ordinals of this subset cannot be μ. But when I considered the following set -

The sequence {ωⁿ} = ω, ω², ... , ωⁿ, ...

- it was hard to discern a limit point other than ω^ω. Aside from what the exact nature of the first uncountable ordinal is chosen to be, there should still be a limit point somewhere before ω^ω. So simply put, what ordinals exist between ω^ω and the sequence I presented?

 

[fresh_42]

@all who like to contribute to this thread please refer to
https://www.physicsforums.com/threads/ordinal-arithmetic.915065/#post-5765141