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

1. Prove that if \(\alpha,\beta,\gamma\) are ordinals, then \[ \alpha \cdot (\beta+\gamma) = \alpha \cdot \beta + \alpha \cdot \gamma, \qquad\qquad \alpha^{\beta + \gamma} = \alpha^\beta \cdot \alpha^\gamma, \qquad \qquad \alpha^{(\beta \cdot \gamma)} = (\alpha^\beta)^\gamma.\]
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 \((L,\leq)\) is a linear ordering, let \(\sigma L\) denote the class of all strictly increasing functions from an ordinal into \((L,\leq)\). Prove that \(\sigma L\) is a set.
5. Prove that there is no function \(f:\sigma L \to (L,\leq)\) 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 \(L\).