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

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 {$\square$} and $♢$. We define ${\Pi }_{\gamma }^1$-indescribability and use the new ${\square }^{\gamma }$-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 ${\Pi }_{\gamma }^1$-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 ${\Pi }_n^1$-indescribable but not ${\Pi }_{n+1}^1$-indescribable, and let $A\subseteq \kappa$ be $n+1$-stationary. Then there are $E_A\subseteq A$ and a ${\square }^n$-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 $♢$ principles; we show that $\gamma$-ineffability is downward absolute to $L$.

Submitted

References in corpus (1)

• Squares and Covering Matrices