Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 5 / iv / a

For uncountable regular $\kappa$, let $\theta =\mathrm{otp}(C_{\zeta })$. Since $C_{\zeta }$ is a club set in $\zeta$, $\mathrm{cf}(\theta )=\mathrm{cf}(\zeta )=\kappa$, so $\theta \ge \kappa$.

If $\theta >\kappa$, its cofinality ensures $\theta >\kappa +\omega$: ordinals strictly between $\kappa$ and $\kappa +\omega$ are successors, and $\kappa +\omega$ has cofinality $\omega$. In the continuous increasing enumeration $c$ of $C_{\zeta }$, take $\gamma =c(\kappa +\omega )$. It is a limit point of $C_{\zeta }$ and has cofinality $\omega <\kappa$. Coherence gives $C_{\gamma }=C_{\zeta }\cap \gamma$, with order type $\kappa +\omega$ and size $\kappa$. This contradicts the size clause for ordinals of cofinality below $\kappa$. ThusThe printed hypotheses need this qualification. At $\kappa =\omega$, choose $C_{\zeta }=\zeta$ for every nonzero countable limit ordinal. These club sets are coherent, and the small-cofinality clause is vacuous. But at $\zeta ={\omega }^2$ their order type is ${\omega }^2$, not $\omega$. The standard square principle includes an order-type bound $\mathrm{otp}(C_{\zeta })\le \kappa$, which also repairs this countable case.