Power sets and countable unions of countable sets (without the Axiom of Choice, of course)

Asaf Karagila

How big are countable unions of countable sets? Not that big, even without the axiom of choice present. But what about their power sets? It turns out that those can be entirely unwieldy. We will explore the various limitations of what ZF can and cannot prove regarding countable unions of countable sets, and what it really means that the power set operation is uncontrollable.

We will only assume a very naive knowledge of set theory as a background.