IDEAL PROJECTIONS AND FORCING PROJECTIONS
IDEAL PROJECTIONS AND FORCING PROJECTIONS
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理想预测和强制预测
DOI：
10.1017/jsl.2013.24
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
M. Zeman
中科院分区：
文献类型：
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作者：
Sean D. Cox;M. Zeman
理想的饱和与Foreman-Magidor-Shelah [10]中的“反链捕获”现象密切相关。我们考虑了几个弱于饱和的反链捕获性质，证明了：（1）如果${\cal I}$是$\omega _2 $上满足平稳反链捕获的正规理想，则存在具有Woodin基数的内模型;（2）对于任意n \in \omega $，相对于大基数是一致的，在n\omega $上存在一个正规理想满足投射反链捕捉，然而，${\cal I}$不是饱和的（甚至不是强的）。这为Foreman在《集合论手册》中的第13个开放性问题提供了否定的答案。
Abstract It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:(1) If ${\cal I}$ is a normal ideal on $\omega _2 $ which satisfies stationary antichain catching, then there is an inner model with a Woodin cardinal;(2) For any $n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal ${\cal I}$ on $\omega _n $ which satisfies projective antichain catching, yet ${\cal I}$ is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).