Exercises for Week 1

1. Prove that every set $y$ has a subset $x \subset y$ with $x \notin y$. Don't use the fact that $y \notin y$ or any other consequences of the Regularity Axiom (which we haven't seen yet.) Hint: consider Russell's paradox and also the separation schema.
2. Define the Kuratowski ordered pairing operation by $(x,y) = \{\{x\},\{x,y\}\}$. Prove that $(a,b) = (c,d)$ holds if and only if $a = c$ and $b = d$.
3. Prove that $\text{Ord}$, the class of ordinals, is transitive and is well-ordered by $\in$.
4. Prove that if $S$ is a set of ordinals, then the union $\alpha = \bigcup S$ is an ordinal, and $\alpha$ is the least upper bound of $S$ in the sense of $\le$ (which was defined for ordinals as $\subset$.)