A trace computation for cofinality strata

For alpha below omega-two, any uncountable cofinality must equal omega-one: cofinality is a cardinal at most the cardinality of alpha, and this is at most aleph-one. If alpha has this cofinality, the stratum theorem with theta equal to alpha and lambda equal to omega says $E_{\omega }^{{\omega }_2}\cap \alpha$ is stationary. This proves the first equality.

Fix such an alpha and an increasing cofinal sequence $(a_{\xi }{)}_{\xi <{\omega }_1}$ in alpha. Recursively define a strictly increasing cofinal sequence c: take $c_0=a_0+1$, at successors take $c_{\xi +1}=\max (c_{\xi },a_{\xi +1})+1$, and at nonzero limits take the supremum of prior values. Countable initial segments remain bounded since alpha has cofinality omega-one. The range is unbounded and closed: a limit point below alpha corresponds to a bounded limit set of indices, whose supremum is below omega-one, and continuity includes that value.

At zero and successor indices the c values are successors and have cofinality one. At nonzero limit indices below omega-one, continuity and strict increase give a countable cofinal sequence with no last point, so the value has cofinality omega. Thus this club avoids $E_{{\omega }_1}^{{\omega }_2}\cap \alpha$. No eligible alpha belongs to its trace, giving the second equality.