## Journal of Symbolic Logic

### Possible Behaviours of the Reflection Ordering of Stationary Sets

Jiri Witzany

#### Abstract

If $S, T$ are stationary subsets of a regular uncountable cardinal $\kappa$, we say that $S$ reflects fully in $T, S < T$, if for almost all $\alpha \in T$ (except a nonstationary set) $S \cap \alpha$ is stationary in $\alpha$. This relation is known to be a well-founded partial ordering. We say that a given poset $P$ is realized by the reflection ordering if there is a maximal antichain $\langle X_p; p \in P\rangle$ of stationary subsets of $\operatorname{Reg}(\kappa)$ so that $\forall p, q \in P \forall S \subseteq X_p, T \subseteq X_q \text{stationary} : (S < T \leftrightarrow p < p q).$ We prove that if $V = L\lbrack\overset{\rightarrow\mathscr{U}}\rbrack, o^\mathscr{U} (\kappa) = \kappa^{++}$, and $P$ is an arbitrary well-founded poset of cardinality $\leq \kappa^+$ then there is a generic extension where $P$ is realized by the reflection ordering on $\kappa$.

#### Article information

Source
J. Symbolic Logic, Volume 60, Issue 2 (1995), 534-547.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744754

Mathematical Reviews number (MathSciNet)
MR1335135

Zentralblatt MATH identifier
0828.03023

JSTOR