Abstract
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
principles. We use these notions to define generalisations of Jensen’s combinatorial
principles ! as !γ and !<γ sequences. We define Π1 -indescribability and show first 1γ1
that in L if γ < κ is an ordinal and κ is Σγ -indescribable but not Πγ -indescribable, and A ⊆ κ is γ-stationary, then there is EA ⊆ A and a !<γ sequence S on κ such that EA is γ-stationary in κ and S avoids EA. This generalises a result of Jensen for γ = 1. As a corollary we also extend the result of Jensen that in L a regular cardinal is stationary reflecting if and only if it is Π1-indescribable by showing that such a κ as above is not γ-reflecting, yielding a different proof of a result appearing in [3]. Thus in L a cardinal is Π1γ-indescribable iff it reflects γ-stationary sets. We define !γ(κ), as stating that there is an unthreadable !γ-sequence at κ; we show this implies that κ is not γ + 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 ♦γ principles.
γ-club and γ-stationary sets, which are closely related to stationary reflection
principles. We use these notions to define generalisations of Jensen’s combinatorial
principles ! as !γ and !<γ sequences. We define Π1 -indescribability and show first 1γ1
that in L if γ < κ is an ordinal and κ is Σγ -indescribable but not Πγ -indescribable, and A ⊆ κ is γ-stationary, then there is EA ⊆ A and a !<γ sequence S on κ such that EA is γ-stationary in κ and S avoids EA. This generalises a result of Jensen for γ = 1. As a corollary we also extend the result of Jensen that in L a regular cardinal is stationary reflecting if and only if it is Π1-indescribable by showing that such a κ as above is not γ-reflecting, yielding a different proof of a result appearing in [3]. Thus in L a cardinal is Π1γ-indescribable iff it reflects γ-stationary sets. We define !γ(κ), as stating that there is an unthreadable !γ-sequence at κ; we show this implies that κ is not γ + 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 ♦γ principles.
| Original language | English |
|---|---|
| Article number | 103272 |
| Number of pages | 52 |
| Journal | Annals of Pure and Applied Logic |
| Volume | 174 |
| Issue number | 7 |
| Early online date | 12 Apr 2023 |
| DOIs | |
| Publication status | Published - 1 Jul 2023 |
Keywords
- Stationary Reflection, Constructibility