Does $\alpha + \beta = \alpha$ imply $\beta \le \aleph_0$

Question

Just like in title, my question is : Does $\alpha + \beta = \alpha$ imply $\beta \le \aleph_0$ where, $\alpha$ and $\beta$ are cardinals?

P.S. I actualy have to prove $\alpha + \beta = \alpha$ $\iff$ $\alpha + \aleph_0 \cdot \beta = \alpha$. Any ideas?

Answer 1

If you mean $\alpha+\beta= \alpha$ as in addition of cardinals, then no. Consider $\alpha=\aleph_2, \beta=\aleph_1$.

Answer 2

$\alpha + \beta = max(\alpha, \beta)$ if one of them is an infinite cardinal, so it's easy to find counterexample.