Which kind product of non-zero number non-zero cardinal numbers yields zero? Let $I$ be a non-empty set. $\kappa_i$ is non-zero cardinal number for all $i \in I$.
If without AC, then $\prod_{i \in I}\kappa_i=0$ seems can be true(despite I still cannot believe it).
But what property should $I$ and $\kappa_i$ have?
Can $\prod_{i \in I}\kappa_i\ne 0$ be proved without AC when $I$ and each $\kappa_i$ all is well-orderable?
Conversely if $I$ is not well-orderable, or if some $\kappa_i$ is not well-orderable, is $\prod_{i \in I}\kappa_i=0$ definitely holds?

If all sets of reals are Lebesgue  measurable, then the product $ \Pi_{x\in \mathbb{R/Q}}x $ , where $\mathbb{R/Q}=\{\{x\in\mathbb{R}|\exists y\in\mathbb{Q}.x+y=z\}|z\in\mathbb{R}\}$, is empty by Vitali's argument. If it is consistent for an inaccessible cardinal to exist, then ZF+"all sets of reals are Lebesgue measurable" + Dependent Choice(A weak form of the axiom of choice) is consistent, according to a theorem of Solovay.
