Generalisations of Stationarity, Closed and Unboundedness, and of Jensen's $\Box$

Abstract

The concepts of closed unbounded (club) and stationary sets are generalised to $\gamma$-club and $\gamma$-stationary sets, which are closely related to stationary reflection. We use these notions to define generalisations of Jensen's combinatorial principles {$\Box$} and $\diamondsuit$. We define $\Pi^1_\gamma$-indescribability and use the new $\Box^{\gamma}$-sequences to extend the result of Jensen that in the constructible universe a regular cardinal is stationary reflecting if and only if it is $\Pi^1_1$-indescribable: we show that in $L$ a cardinal is $\Pi^{1}_{\gamma} $-indescribable iff it reflects $\gamma$-stationary sets. More particularly (stating only the special case of $n$ finite): Theorem $(V=L)$ Let $n<\omega$ and $\kappa$ be $\Pi^{1}_{n}$-indescribable but not $\Pi^1_{n+1}$-indescribable, and let $A\subseteq\kappa$ be $n+1$-stationary. Then there are $E_{A}\subseteq A$ and a $\Box^{n}$-sequence $S$ on $\kappa$ such that $E_{A}$ is $n+1$-stationary in $\kappa$ and $S$ avoids $E_{A}$. Thus $\kappa$ is not $n+1$-reflecting. Certain assumptions on the $\gamma$-club filter allow us to prove that $\gamma$-stationarity is downwards absolute to $L$, and allows for splitting of $\gamma$-stationary sets. We define $\gamma$-ineffability, and look into the relation between $\gamma$-ineffability and various $\diamondsuit$ principles; we show that $\gamma$-ineffability is downward absolute to $L$.