TY - JOUR

T1 - Local saturation of the non-stationary ideal over Pκ λ

AU - Usuba, Toshimichi

PY - 2007/11/1

Y1 - 2007/11/1

N2 - Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of Pκ λ such that NSκ λ | S, the ideal generated by the non-stationary ideal NSκ λ over Pκ λ together with Pκ λ {set minus} S, is λ+-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH). We also show that in our model we can make NSκ λ | S (κ, λ) λ+-saturated, where S (κ, λ) is the set of all x ∈ Pκ λ such that ot (x), the order type of x, is a regular cardinal and x is stationary in sup (x). Furthermore we construct a model where NSκ λ | S (κ, λ) is κ+-saturated but GCH fails. We show that if S {set minus} S (κ, λ) is stationary in Pκ λ, then S can be split into λ many disjoint stationary subsets.

AB - Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of Pκ λ such that NSκ λ | S, the ideal generated by the non-stationary ideal NSκ λ over Pκ λ together with Pκ λ {set minus} S, is λ+-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH). We also show that in our model we can make NSκ λ | S (κ, λ) λ+-saturated, where S (κ, λ) is the set of all x ∈ Pκ λ such that ot (x), the order type of x, is a regular cardinal and x is stationary in sup (x). Furthermore we construct a model where NSκ λ | S (κ, λ) is κ+-saturated but GCH fails. We show that if S {set minus} S (κ, λ) is stationary in Pκ λ, then S can be split into λ many disjoint stationary subsets.

KW - Club-shooting

KW - GCH

KW - Non-stationary ideal

KW - P λ

KW - Saturated ideal

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U2 - 10.1016/j.apal.2007.08.002

DO - 10.1016/j.apal.2007.08.002

M3 - Article

AN - SCOPUS:35348992040

VL - 149

SP - 100

EP - 123

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

SN - 0168-0072

IS - 1-3

ER -