I'm trying to prove that the set of limit ordinals in $\omega_1$, call it $L$, is unbounded. This result is intuitively clear, but explicitly showing this isn't as intuitive.
Toward a contradiction, suppose that $L$ is bounded. Then I know a couple of things follow:
- I know that countable sets are bounded, but are bounded sets countable?
- Because $L$ is the set of limit ordinals, we cannot suppose $\alpha = \sup L$ and use $\alpha +1$ as a contradiction
Any advice on how to do this? Is a contradiction best or is there a better method?
Your idea of just using $\alpha+1$ works with a little modification: if $\alpha$ is a countable ordinal, then $\alpha+\omega$ is a countable limit ordinal.
I wouldn’t phrase this as a proof by contradiction but as a direct proof that the limit ordinals less than $\omega_1$ are unbounded.