Exercises for Week 1
Due Wednesday, October 9
- 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.
- 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 $b = d$.
- Prove that $\text{Ord}$, the class of ordinals, is transitive and is well-ordered by $\in$.
- 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 $\subset$.)