Can Aleph Numbers be multiplied?

Question

i.e., does it make sense to say something like $(2 * \aleph_0) > \aleph_0$ ?

The original question I was thinking about is:

if A = $\mathbb{Z}$ and B = {the set of even integers} is it correct to say that |A| = 0.5|A| + 0.5|B| ?

Answer 1

The aleph numbers can certainly be multiplies, but not to give the results you mention. In fact, $2\aleph_0=\aleph_0$ and there is no such thing as fractional alephs so $0.5|A|$ is meaningless.

For the sets $A$ and $B$ you mention it is very easy to establish a bijection and thus $|A|=|B|$.