My introduction into Axiom of Choice has been kind of confusing (Zorn's lemma) for the start, so it took me some time to realize it's nothing but to say The non-empty product of non-empty sets is non-empty. I still find it quite puzzling that this doesn't follow from the plain ZF axioms, but still ...
Just to understand the nature of infinite Cartesian products better now, I has asked myself the following: Does the Cartesian product get "bigger" the more non-empty sets I take, as it is with finite products ($|\{1,2\}|\leq |\{1,2\}^2| $)? To put it more correctly
Let $\{A_i\}_{i \in I}$ be a family of non-empty sets and $J \subset I$ a subset of the index set $I$. Does my intuition hold and the following is correct?$\left|\prod_{j\in J} A_j \right| \leq \left|\prod_{i\in I} A_i\right|$
As I see it, if this was true, Axiom of Choice follows (let $J$ be finite $\Rightarrow$ LHS is not empty), but is the statement equivalent to the Axiom of Choice? If so, how would one prove that?