How large is a countable union of countable sets? Assuming ZFC, the answer is simple: it’s countable, and in particular its power set has the same size as the real numbers. Assuming only ZF it turns out that we cannot say a whole lot about countable unions of countable sets, and we can say even less about their power sets. We will present some theorems regarding what we can and cannot determine about countable unions of countable sets and about their power sets, and see that ZF is even weaker than ZFC when it comes to saying something
concrete about power sets of infinite sets.