Recreational problems in set theory?

Question

Most areas of maths that I can think of have a number of fun, recreational problems that come under their category. Nothing deep: number theoretic stuff in olympiads, integrals, limits, products, series in real/complex analysis, colouring/construction problems in graph theory, cool little existence problems in group theory, the list goes on.

Set theory has always felt solely research bent to me - most of the related questions posted on here seem quite deep, or arising from serious study.

Are there any "fun" set-theoretic problems out there? If so it would be interesting to gather a little collection here.

Answer 1

I think it's kinda fun to see how so many things elegantly follow from definitions or axioms:

• The axiom of regularity says that every nonempty set $x$ has an element $y$ that is disjoint from $x$: $$ \forall x : (x\neq \emptyset \rightarrow \exists y\in x : (y\cap x= \emptyset )) $$ Conclude that:
  • $a \notin a$
  • $a \notin b$ or $b \notin a$
  • There is no infinite sequence $a_1 \ni a_2 \ni a_3 \dots$

• An ordinal number is a set $\alpha$ that is

  • transitive, that is, for every $x \in \alpha$, we have $x \subseteq \alpha$, and
  • totally ordered w.r.t. set inclusion, that is, for every $x, y \in \alpha$ we have $x \subseteq y$ or $y \subseteq x$.

  Show that for any $\beta \in \alpha$, $\beta$ is an ordinal number as well.

  I find this one especially cute since

  to show that $\beta$ is transitive, you need the fact that $\alpha$ is ordered w.r.t. set includion, and vice versa.

Answer 2

Fast growing hierarchy? If that doesn't count as set theory, you can try to define large countable ordinals, or just large cardinals. And don't forget Hamkins, the king of recreational set theory!