Is a number devided by two different sizes of infinite is equal?

Question

Let x be any number, Is $x / \aleph_0 = x / \aleph_1 = 0$? Does the size of the infinity matter? Thanks.

Answer 1

As stated in the comments, you need to define the division operation you are talking about. The only reasonable definition of division for cardinal numbers that I can think of would put: $$ \frac{\kappa}{\mu} = \sup \{ \alpha \mid \alpha\mu \le \kappa\} $$

with the proviso that $\mu \neq 0$. Here $\alpha$ ranges over cardinals and $\alpha\mu$ is cardinal multiplication: $\alpha\mu = |\alpha \times \mu|$ (the cardinal of the cartesian product). This agrees with division with rounding down for finite cardinals. It is not very interesting for infinite cardinals: for infinite $\kappa$ and $\mu$ we will have

$$ \frac{\kappa}{\mu} = \left\{ \begin{array}{ll} 0 & \mbox{if $\kappa < \mu$}\\ \kappa & \mbox{otherwise} \end{array} \right. $$