The non-compactness of square
The non-compactness of square
复制标题
正方形的非紧性
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
M. Magidor
中科院分区：
文献类型：
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作者：
J. Cummings;M. Foreman;M. Magidor
本文证明了两个定理。第一个是对每个n都有，但没有。这是通过仔细折叠超紧基数并将平方序列添加到每个ωn来完成的。证明的关键在于，在所得到的模型中，<$ω+1 <$cof（ω）的每个静止子集都反映到一个共尾性为ω1的序数，也就是说它与这样一个序数有静止交集。这一结果与文献[3]中所给出的正方形的紧性性质不同。在这篇文章中，我们证明了如果在每个ωn处都有平方，那么在ωk，k > 1的共尾点上有一个平方型序列.特别地，在共尾性大于ω1的点处，存在可数共尾性的强非反射平稳点集。第二个结果回答了Džamonja的一个问题，通过证明在超紧基数上不可能有类平方序列，其中“类平方”意味着用它们在不可数共尾的所有点上都是平稳的要求来取代共尾集是封闭和无界的要求。
This note proves two theorems. The first is that it is consistent to have for every n, but not have . This is done by carefully collapsing a supercompact cardinal and adding square sequences to each ωn. The crux of the proof is that in the resulting model every stationary subset of ℵω+1 ⋂ cof(ω) reflects to an ordinal of cofinality ω1, that is to say it has stationary intersection with such an ordinal. This result contrasts with compactness properties of square shown in [3]. In that paper it is shown that if one has square at every ωn, then there is a square type sequence on the points of cofinality ωk, k > 1 in ℵω+1. In particular at points of cofinality greater than ω1 there is a strongly non-reflecting stationary set of points of countable cofinality. The second result answers a question of Džamonja, by showing that there can be no squarelike sequence above a supercompact cardinal, where “squarelike” means that one replaces the requirement that the cofinal sets be closed and unbounded by the requirement that they be stationary at all points of uncountable cofinality.