A quasi-lower bound on the consistency strength of PFA
A quasi-lower bound on the consistency strength of PFA
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DOI：
10.1090/s0002-9947-2014-05955-2
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发表时间：
2014-03
影响因子：
1.3
通讯作者：
S. Friedman;P. Holy
中科院分区：
文献类型：
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作者：
S. Friedman;P. Holy
一个长期存在的问题是，超紧性是否提供了一个下界的一致性强度的本征强迫公理（PFA）。在这篇文章中，我们建立了一个拟下界，证明了存在一个具有适当的次紧基数类的模型，使得PFA（实际上是PFA对（2 <$0）+-联结强迫成立的较弱的陈述）在其所有的适当强迫扩张中都失败。尼曼得到这样的结果假设存在的“精细结构”模型包含非常大的基数，但存在这样的模型仍然开放。我们表明，尼曼的论点通过类似的概念“L型”模型，并建立存在的Llike模型包含非常大的基数。所需的主要技术成果是在非常大的红衣主教存在的情况下本地俱乐部凝聚与可接受性的兼容性，这一结果构成了外部模型计划的进一步进展。核心模型计划（由詹森发起，参见斯蒂尔的[16]调查）在建立集合论陈述的一致性强度的下限方面取得了相当大的成功，达到了Woodin基数的水平。但本征强迫公理（PFA）的相容性强度被证明是超紧基数的相容性强度，目前还没有可用的核心模型理论。因此，值得考虑的拟下界的一致性强度的PFA和本文的主要结果是，一个适当的类的次紧基数作为这样的拟下界：定理1。假设一个适当的次紧基数类的相容性，则存在一个适当的次紧基数类是相容的，但PFA（甚至限于（2 <$0）+-连接的偏序集）在论域的任何适当扩张中都不成立。准下界究竟是什么意思？必要的成分是·所需的集合论原理φ，我们希望获得2000年数学主题分类的准下限结果。03E35 03E55 03E57
A long-standing open question is whether supercompactness provides a lower bound on the consistency strength of the Proper Forcing Axiom (PFA). In this article we establish a quasi lower bound by showing that there is a model with a proper class of subcompact cardinals such that PFA (indeed the weaker statement that PFA holds for (2א0)+-linked forcings) fails in all of its proper forcing extensions. Neeman obtained such a result assuming the existence of “fine structural” models containing very large cardinals, however the existence of such models remains open. We show that Neeman’s arguments go through for a similar notion of “L-like” model and establish the existence of Llike models containing very large cardinals. The main technical result needed is the compatibility of Local Club Condensation with Acceptability in the presence of very large cardinals, a result which constitutes further progress in the outer model programme. The core model programme (initiated by Jensen, see Steel’s [16] for a survey) has had considerable success in establishing lower bounds on the consistency strength of set-theoretic statements, up to the level of Woodin cardinals. But the consistency strength of the Proper Forcing Axiom (PFA) is conjectured to be that of a supercompact cardinal, for which no core model theory is currently available. It is therefore worthwhile to consider quasi lower bounds on the consistency strength of PFA and the main result of this paper is that a proper class of subcompact cardinals serves as such a quasi lower bound: Theorem 1. Assuming the consistency of a proper class of subcompact cardinals, it is consistent that there is a proper class of subcompact cardinals, but PFA (even restricted to posets which are (2א0)+-linked) holds in no proper extension of the universe. What exactly is meant by a quasi lower bound? The necessary ingredients are • the desired set-theoretic principle φ for which we want to obtain a quasi-lower bound result 2000 Mathematics Subject Classification. 03E35, 03E55, 03E57.