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