The following statement can be proven using elementary submodels and sufficiently generic conditions:

"If $S \subseteq cof(<\kappa) \cap \kappa^+$ is stationary, and $\kappa^{<\kappa} =\kappa$, then the stationarity of $S$ is preserved by $\kappa$-closed forcing."

If we just assume $\kappa$ is regular, do we need the cardinal arithmetic?


No: your cardinal arithmetic assumption can be dropped when $\kappa$ is regular.

This follows from $I[\lambda]$ analysis.

$S \subseteq \lambda \cap \mathrm{cof}(\kappa)$ (where $\kappa < \lambda$ are regular) is said to be $\textit{in $I[\lambda]$}$ if there is a sequence of sets $\langle a_i \mid i < \lambda \rangle$ and a club $C\subseteq \lambda$ such that every $\delta \in S \cap C$ is approachable w.r.t $\vec{a}$.

$\delta \in S$ is said to be approachable w.r.t. $\vec{a}$ when there is an unbounded subset $A \subseteq \delta$ of order-type $\kappa$ such that $\{ A \cap \alpha \mid \alpha < \delta \} \subseteq \{ a_i \mid i < \delta \}$.

Shelah proved the following:

  1. $S \subseteq \lambda \cap \mathrm{cof}(\kappa)$ is indestructible by $\kappa^+$-closed forcings if and only if, for every large regular $\theta >> \lambda$ and every $x \in H(\theta)$, there are an elementary submodel $M \ni x$ and $\delta \in S$, and an unbounded $A \subseteq \delta$ such that:

    • $\delta= M\cap \lambda$,
    • $\mathrm{otp}(A)= \kappa$,
    • $\{ A \cap \alpha \mid \alpha < \delta \} \subseteq M$.
  2. If $\kappa$ are regular, $\kappa^+ \cap \mathrm{cof}(< \kappa) \in I[\kappa^+]$.

By 1, it is easy to check that if a stationary set $S \subseteq \kappa^+ \cap \mathrm{cof}(\nu)$($\nu< \kappa$: regular) is in $I[\kappa^+]$ then $S$ is indestructible by $\nu^+$-closed forcings, and hence is indestructible by $\kappa$-closed forcings.

By Shelah's result 2, $I[\kappa^+] \restriction \mathrm{cof}(< \kappa)$ is improper, so every stationary set $S \subseteq \kappa^+ \cap \mathrm{cof}(<\kappa)$ is preserved by $\kappa$-closed forcings, without any cardinal arithmetic.

For proofs, see Cummings' article ``Notes on Singular Cardinal Combinatorics.''


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.