I’m trying to understand ordinal arithmetic. If one had an ordered list of the some subset of countable ordinal numbers, what order would the following 6 countably infinite ordinals be in? If the following order is not correct, what is correct order and why is that the correct order?
$$omega;<; omega^2 ;<; 2^omega ;<; omega^omega ;<; {^omega}2 ;<;{^omega} omega$$
I know $epsilon_0 = {^omega} omega$ is the largest, but is still countable, but I’m not sure where the powers of $2$ fit in versus the powers of $omega$.
I understand why $omega^2$ or $omega^n$ for any finite value of $n$ needs to be countable. But, why does $omega^omega$ need to be countable? For cardinal numbers, $2^{aleph_0}$ is uncountably infinite. Presumably, there would be some contradiction in mathematics if any finite ordinal arithmetic equation involving $omega$ generated an uncountable infinity.