Give an example of a limit ordinal which is not a cardinal and show that it is not a cardinal.

Question

By Definition 9.1.1. in my textbook it says that a cardinal is an ordinal such that for every $\beta \in \kappa$ there is no one-to-one function $f:\kappa \rightarrow \beta$.

I found out that $\omega^2$ is a limit ordinal, because it is not a successor ordinal. Next, do I have to show that there exists a one-to-one function $f:\omega^2 \rightarrow \omega$ by contradiction to show that it is not a cardinal? Or what do I do for my next step?

Any help is appreciated (I have already looked around at the site for similar problems).

Answer 1

$\omega+\omega$ is a limit ordinal which is not a cardinal. You can define a one to one function $f:\omega+\omega\to\omega$ by $n\to 2n$ and $\omega+n\to 2n+1$ for all $n\in\mathbb{N}$.