Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 24 / 3 / a

The diagonal intersection of the family isMembership at $\beta$ only tests the sets indexed below $\beta$, rather than every set of the family. At a regular uncountable $\delta$, a diagonal intersection of club sets is again a club set.

Fodor lemma says that if $\lambda$ is a regular uncountable cardinal, $S\subseteq \lambda$ is stationary, and $f:S\to \lambda$ is regressive, thenHere a regressive function satisfies $f(\alpha )<\alpha$ at every nonzero $\alpha \in S$; removing $0$ has no effect on stationarity. Both regularity and stationarity are part of the hypotheses.

The club principle $♣$ asserts that there is a sequence $⟨A_{\delta }:\delta \in \mathrm{Lim}({\omega }_1)⟩$ such that $A_{\delta }\subseteq \delta$ has order type $\omega$, is cofinal in $\delta$, andIt guesses a cofinal countable subset of an uncountable set, with inclusion as the guessing requirement.