scholarly journals PFA and precipitousness of the nonstationary ideal

2017 ◽  
Vol 146 (2) ◽  
pp. 791-802
Author(s):  
Boban Veličković
Keyword(s):  
Nonstationary Ideal

Increasing u2 by a stationary set preserving forcing

2009 ◽  
Vol 74 (1) ◽  
pp. 187-200
Author(s):  
Benjamin Claverie ◽  
Ralf Schindler
Keyword(s):  
Regular Cardinal ◽  
Nonstationary Ideal ◽  
Stationary Set ◽  
Precipitous Ideal ◽  
Countable Structure

AbstractWe show that if I is a precipitous ideal on ω1 and if θ > ω1 is a regular cardinal, then there is a forcing ℙ = ℙ(I, θ) which preserves the stationarity of all I-positive sets such that in Vℙ, ⟨Hθ; ∈, I⟩ is a generic iterate of a countable structure ⟨M; ∈, Ī⟩. This shows that if the nonstationary ideal on ω1 is precipitous and exists, then there is a stationary set preserving forcing which increases . Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω1 is precipitous, then .

scholarly journals Cofinality of the nonstationary ideal

2005 ◽  
Vol 357 (12) ◽  
pp. 4813-4837 ◽  
Author(s):  
Pierre Matet ◽  
Andrzej Rosłanowski ◽  
Saharon Shelah
Keyword(s):  
Nonstationary Ideal

On precipitousness of the nonstationary ideal over a supercompact

1986 ◽  
Vol 51 (3) ◽  
pp. 648-662 ◽  
Author(s):  
Moti Gitik
Keyword(s):  
General Theorem ◽  
Supercompact Cardinal ◽  
Generic Extension ◽  
Inner Model ◽  
Nonstationary Ideal

Namba [N] proved that the nonstationary ideal over a measurable (NSκ) cannot be κ+-saturated. Baumgartner, Taylor and Wagon [BTW] asked if it is possible for NSκ to be precipitous over a measurable κ. A model with this property was constructed by the author, and shortly after Foreman, Magidor and Shelah [FMS] proved a general theorem that after collapsing of a supercompact or even a superstrong to the successor of κ, NSκ became precipitous. This theorem implies that it is possible to have the nonstationary ideal precipitous over even a supercompact cardinal. Just start with a supercompact κ and a superstrong λ > κ. Make supercompactness of κ indistractible as in [L] and then collapse λ to be κ+.The aim of our paper is to show that the existence of a supercompact cardinal alone already implies the consistency of the nonstationary ideal precipitous over a supercompact. The proof gives also the following: if κ is a λ-supercompact for λ ≥ (2κ)+, then there exists a generic extension in which κ is λ-supercompact and NSκ is precipitous. Thus, for a model with NSκ precipitous over a measurable we need a (2κ)+-supercompact cardinal κ. Jech [J] proved that the precipitous of NSκ over a measurable κ implies the existence of an inner model with o(κ) = κ+ + 1. In §3 we improve this result a little by showing that the above assumption implies an inner model with a repeat point.The paper is organized as follows. In §1 some preliminary facts are proved. The model with NSκ precipitous over a supercompact is constructed in §2.

scholarly journals On the universality of the nonstationary ideal

2018 ◽  
Vol 64 (1-2) ◽  
pp. 103-117 ◽  
Author(s):  
Sean D. Cox
Keyword(s):  
Nonstationary Ideal

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.

Some results on the nonstationary ideal

1995 ◽  
Vol 92 (1-3) ◽  
pp. 61-112 ◽  
Author(s):  
Moti Gitik
Keyword(s):  
Nonstationary Ideal
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