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

As a product of cardinals, yes:
$$2^{\aleph_0} \leq \prod_{0 < n < \omega} n \leq {\aleph_0}^{\aleph_0} \leq 2^{\aleph_0 \cdot \aleph_0} = 2^{\aleph_0}$$
As a product of ordinals, no:
$$\prod_{0 < n < \omega} n \leq \prod_{0 < n < \omega} \omega = {\omega}^{\omega}$$ 
but the ordinal ${\omega}^{\omega}$ is countable.
