Do any three problems. Click here to return to the main course web page.

1. Prove that if $\alpha ,\beta ,\gamma$ are ordinals, then $$\alpha \cdot (\beta +\gamma )=\alpha \cdot \beta +\alpha \cdot \gamma ,{\alpha }^{\beta +\gamma }={\alpha }^{\beta }\cdot {\alpha }^{\gamma },{\alpha }^{(\beta \cdot \gamma )}=({\alpha }^{\beta }{)}^{\gamma }.$$
2. Prove that a positive ordinal $\alpha$ is closed under ordinal addition if and only if $\alpha ={\omega }^{\xi }$ for some ordinal $\xi$. Such ordinals are said to be (additively) indecomposible. Find and prove a similar characterization of when a positive ordinal is closed under multiplication.
3. Show that there is a well order in $(V_{\omega \cdot 2},\in )$ which is not isomorphic to an ordinal. In particular, $(V_{\omega \cdot 2},\in )$ does not satisfy the Collection Scheme. Remark: $V_{\omega \cdot 2}$ contains most objects encountered in mathematics.
4. If $(L,\le )$ is a linear ordering, let $\sigma L$ denote the class of all strictly increasing functions from an ordinal into $(L,\le )$. Prove that $\sigma L$ is a set.
5. Prove that there is no function $f:\sigma L\to (L,\le )$ such that if $s,t\in \sigma L$ and $s$ is a proper restriction of $t$, then $f(s)<f(t)$. Hint: suppose for contradiction that this is not the case and use the Recursion Theorem to define a strictly increasing function from $\mathbf{ON}$ into $L$.