Recreational problems in set theory?

Solution 1:

Here's a good one: find an explicit bijection between two intervals [0,1] and [0,1).

Solution 2:

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.

Solution 3:

Whenever you feel like using the axiom of choice, first ask yourself whether you could build a choice function instead. This yields fun problems more often than not.