Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 3 / i

For a nonzero limit ordinal $\delta$, a subset $C\subseteq \delta$ is a club set if it is unbounded in $\delta$ and contains every one of its limit points below $\delta$. A subset $S\subseteq \delta$ is stationary ifThis defines stationarity in a limit ordinal. The regular uncountable case is the usual stationary set setting; results such as Fodor lemma require that additional hypothesis, rather than an arbitrary limit ordinal.

A set-theoretic tree is a partial order for which the strict predecessors of every node are well-ordered by the tree order. The height of a node is that predecessor order type, and $T_{\alpha }$ is the set of nodes of height $\alpha$. Under the ordinary height-and-width definition, a kappa-tree satisfies

For an arbitrary cardinal in this definition one must distinguish it from additional conventions such as being well-pruned, normal or splitting. Splitting means that every node has two incompatible extensions; it is not implied by the height-and-width clauses. This distinction is material in Question 5(ii)(a).

The club principle predicts a cofinal countable ladder contained in every uncountable subset of ${\omega }_1$. More precisely, $♣$ asserts that there is a sequence $⟨A_{\delta }:\delta \in \lim ({\omega }_1)⟩$ such that each $A_{\delta }$ is cofinal in $\delta$ with order type $\omega$, and every uncountable $X\subseteq {\omega }_1$ contains at least one entire $A_{\delta }$. It predicts a countable ladder contained in $X$, rather than predicting $X\cap \delta$ exactly. The latter is the stronger kind of guessing in the diamond principle.