Stationary subsets and stabilizers of restrictive Menger P-algebras of multiplace functions

2000 ◽  
Vol 44 (1-2) ◽  
pp. 129-142 ◽  
Author(s):  
Valentin S. Trokhimenko
Keyword(s):  
Stationary Subsets

Some Stationary Subsets of (λ)

1988 ◽  
Vol 102 (4) ◽  
pp. 1000 ◽  
Author(s):  
Hans-Dieter Donder ◽  
Peter Koepke ◽  
Jean-Pierre Levinski
Keyword(s):  
Stationary Subsets

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

2017 ◽  
Vol 82 (3) ◽  
pp. 1106-1131 ◽  
Author(s):  
PHILIPP LÜCKE ◽  
RALF SCHINDLER ◽  
PHILIPP SCHLICHT
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
Woodin Cardinals ◽  
Perfect Set ◽  
Nonstationary Ideal ◽  
Definable Sets ◽  
Image Position ◽  
Stationary Subsets ◽  
Woodin Cardinal ◽  
Generic Ultrapowers

AbstractWe study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

scholarly journals Possible behaviours of the reflection ordering of stationary sets

1995 ◽  
Vol 60 (2) ◽  
pp. 534-547 ◽  
Author(s):  
Jiří Witzany
Keyword(s):  
Partial Ordering ◽  
Generic Extension ◽  
Maximal Antichain ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Stationary Subsets ◽  
Almost All

AbstractIf S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T, S < T, if for almost all α ∈ T (except a nonstationary set) S ∩ α stationary in α. 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 〈Xp: p ∈ P〉 of stationary subsets of Reg(κ) so thatWe prove that if , and P is an arbitrary well-founded poset of cardinality ≤ κ+ then there is a generic extension where P is realized by the reflection ordering on κ.

Splitting P κ λ into Stationary Subsets

1988 ◽  
Vol 53 (2) ◽  
pp. 385 ◽  
Author(s):  
Yo Matsubara
Keyword(s):  
Stationary Subsets

GAMES OF LENGTH ω1

2007 ◽  
Vol 07 (01) ◽  
pp. 83-124 ◽  
Author(s):  
ITAY NEEMAN
Keyword(s):  
Minimal Model ◽  
Woodin Cardinals ◽  
Stationary Subsets

We prove determinacy for open length ω1 games. Going further we introduce, and prove determinacy for, a stronger class of games of length ω1, with payoff conditions involving the entire run, the club filter on ω1, and a sequence of ω1 disjoint stationary subsets of ω1. The determinacy proofs use an iterable model with a class of indiscernible Woodin cardinals, and we show that the games precisely capture the theory of the minimal model for this assumption.

Splitting Pκλ into stationary subsets

1988 ◽  
Vol 53 (2) ◽  
pp. 385-389
Author(s):  
Yo Matsubara
Keyword(s):  
Inaccessible Cardinal ◽  
Nonstationary Ideal ◽  
Stationary Subsets

AbstractWe show that if κ is an inaccessible cardinal then Pκλ splits into λ<κ many disjoint stationary subsets. We also show that if Pκλ carries a strongly saturated ideal then the nonstationary ideal cannot be λ+-saturated.

Stationary subsets of [ℵω]<ωn

1993 ◽  
Vol 58 (4) ◽  
pp. 1201-1218 ◽  
Author(s):  
Kecheng Liu
Keyword(s):  
Large Cardinals ◽  
Image Position ◽  
Stationary Subsets

AbstractIn this paper, assuming large cardinals, we prove the consistency of the following:Let n ∈ ω and k1, k2 ≤ n. Let f: ω → {k1, k2} be such that for all n1 < n2 ∈ f−1{k1},n2 − n1 ≥ 4. Then the setis stationary in The above is equivalent to the statement that for any structure on on ℵω, there is ≺ A such that ∣∣ = ωn and for all m > n, cf( ∩ ωm) = ωf(m).

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