What is the product of all nonzero, finite cardinals? To be specific, why does the following equality hold?
$$
\prod_{0\lt n\lt\omega}n=2^{\aleph_0}
$$

If $\displaystyle f\in\prod_{n\in\omega} 2$ then $f(n)\in\{0,1\}$, and in particular for $n>1$ we have that $f(n)\in n$. Therefore this is a proper subset of $\displaystyle f\in\prod_{0<n<\omega} n$, therefore the cardinality is at least continuum.
On the other hand $\omega^\omega$ has cardinality continuum, and the same argument shows that the product is a  subset of $\displaystyle\prod_{n\in\omega}\omega$
