6
$\begingroup$

Assume $\kappa > \omega$ is a cardinal of uncountable cofinality and $S$ is a stationary set in $\kappa$. If $\alpha < \kappa$ is a successor ordinal then the set $S+\alpha=\{\sigma+\alpha : \sigma \in S\}$ is not stationary in $\kappa$ because it does not intersect the subset of limit ordinals in $\kappa$ (which is club in $\kappa$). Is $S + \alpha$ stationary in $\kappa$ if $\alpha < \kappa$ is a limit ordinal? What about the set $\{\sigma + \sigma : \sigma \in S\}$?

  • 3
    If I recall correctly, the set of limits of limits is a club. Then if $\alpha=\omega,\ S+\alpha$ does not intersect this club.2011-06-21

1 Answers 1

5

If $\kappa$ is regular and uncountable, then no set of the form $S = A + \alpha$ can be stationary (for any $A \subseteq \kappa$ and $\alpha \in \kappa$) by Fodor's lemma. The function which sends each $\beta \in S$ to the least $\gamma$ such that $\gamma + \alpha = \beta$ is regressive.

A similar argument shows that no set of the form $\{\alpha + \alpha : \alpha \in A\}$ is stationary (again, for regular $\kappa$).

  • 0
    @Brian M. Scott: It's true that in most formulations of Fodor's lemma the regularity of $\kappa$ is assumed. However, the following proposition is also true: if $\alpha$ is a limit ordinal with uncountable cofinality and $f: S \to \alpha$ is a regressive function from a stationary set $S \subset \alpha$, then the set $f^{-1}[0,\beta)$ is stationary in $\alpha$ for some \beta < \alpha.2011-06-23