How do you prove the following transfinite cardinal addition?:
$ \alpha + \beta = \max(\alpha,\beta)$?
And as the consequence, $\alpha + \alpha = \alpha$ where $\alpha$ and $\beta$ are transfinite cardinal numbers?
To show that $\alpha + \beta = \beta$ whenever $\alpha \leq \beta$ are cardinals and $\beta$ is infinite, one often goes through the following sequence of deductions:
The "trick" for the first part is to construct a well-ordering on $\beta \times \beta$ (the cartesian product) which has order-type $\beta$.
Since there is a well-ordering on $\beta \times \beta$ of order-type $\beta$, it follows that the sets $\beta \times \beta$ and $\beta$ have the same cardinality. As we define the cardinal product $\beta \cdot \beta$ to be $| \beta \times \beta |$, we are done.
How would you prove $\aleph_0 + \aleph_0 = \aleph_0$? If you do that in the obvious way, can you think about how to generalize it to arbitrary cardinals $\alpha$? (It might help to think about Cantor normal form for the latter, but not necessary). Then if you can do that, can you see how you'd get $\alpha + \beta = \max(\alpha, \beta)$?