In the chapter on stationary sets in the handbook of set theory, for $\kappa$ a regular uncountable cardinal and $\lambda < \kappa$ regular, Jech defines $$E_{\lambda}^{\kappa} = \{\alpha < \kappa \mid \operatorname{cf}(\alpha)=\lambda \}.$$ Now, it seems that $\lambda$ itself should be a member of $E_{\lambda}^{\kappa}$ as it satisfies $\operatorname{cf}(\lambda)=\lambda<\kappa$. But a paragraph later, it is claimed that "$\bigcup_\lambda E_{\lambda}^{\kappa}$ is the set of all singular limit ordinals less than $\kappa$", so $\lambda$ is not there.

Am I missing something?


1 Answer 1


You're correct about this. $\lambda$ is in fact of cofinality $\lambda$, so it should be in $E^\kappa_\lambda$. But any other ordinal in $E^\kappa_\lambda$ is singular of cofinality $\lambda$, and removing one element from a stationary set is entirely inconsequential.

  • 1
    $\begingroup$ yes, sure, I was just wondering about the convention since the definition and the claim seem to contradict each other... $\endgroup$
    – Ur Ya'ar
    May 17, 2017 at 9:50

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