Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 100-123 |
| Number of pages | 24 |
| Journal | Annals of Pure and Applied Logic |
| Volume | 149 |
| Issue number | 1-3 |
| DOIs | |
| Publication status | Published - 2007 Nov |
| Externally published | Yes |
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Keywords
- Club-shooting
- GCH
- Non-stationary ideal
- P λ
- Saturated ideal
ASJC Scopus subject areas
- Logic
Cite this
Local saturation of the non-stationary ideal over Pκ λ. / Usuba, Toshimichi.
In: Annals of Pure and Applied Logic, Vol. 149, No. 1-3, 11.2007, p. 100-123.Research output: Contribution to journal › Article
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TY - JOUR
T1 - Local saturation of the non-stationary ideal over Pκ λ
AU - Usuba, Toshimichi
PY - 2007/11
Y1 - 2007/11
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 -