I'd like to know about the examples of ordinal numbers between $\displaystyle \omega_1 (=\aleph_1) $ and $\displaystyle \omega_2 (=\aleph_2) $. In the case between $\displaystyle \omega (=\aleph_0) $ and $\displaystyle \omega_1 $, we know that we have $\displaystyle \omega +1 $, ..., $\displaystyle \omega \times 2 $, ..., $\displaystyle \omega \times 3 $, ..., $\displaystyle \omega^2 $, ..., $\displaystyle \omega^3 $, ..., $\displaystyle \omega^\omega $, ..., $\displaystyle (\omega^\omega) \times 2 $, ..., $\displaystyle \omega^{(\omega +1)} $, ..., $\displaystyle \omega^{(\omega^\omega)} $, ..., $\displaystyle \omega^{(\omega^{(\omega^\omega)})} $, ..., $\displaystyle \varepsilon_0, ..., \varepsilon_1, ... $, and so on, using Cantor Normal Form. But I don't know what kind of ordinal numbers between $\displaystyle \omega_1 $ and $\displaystyle \omega_2 $. If someone know any ordinal numbers (different from the kinds listed above), could you tell me some?

Thank you for viewing this thread and I'm looking forward to your replies!

Misako Kawasoe