Iterated Cohen extensions and Souslin's problem*-猫眼课题宝

迭代科恩扩展和苏斯林问题*

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1971

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R. Solovay;S. Tennenbaum

通过以下性质列表，我们可以在序同构的意义上刻画实数直线：\(R\)是序完备的、序稠密的，没有首元素或末元素，并且包含一个可数稠密子集。（首先证明可数稠密子集与有理数集\(Q\)是序同构的，然后证明这个有序集与其稠密子集的戴德金完备化是同构的。）苏斯林提出了一个问题，即“可数稠密子集”这个条件是否可以用以下结果来替代[15]：（\(*\)）每一个互不相交的非空开区间族是可数的。我们用\(SH\)（苏斯林假设）来表示以下命题：每一个满足（\(*\)）的序完备、序稠密的线性有序集都包含一个可数稠密子集。我们用\(ZFC\)来表示策梅洛 - 弗兰克尔集合论（包括选择公理）。在[16]中，滕嫩鲍姆构造了\(ZFC\)的一些模型，在其中\(SH\)不成立。此外，在其中一个模型中连续统假设（\(CH\)）不成立，而在另一个模型中，广义连续统假设（\(GCH\)）成立。因此\(SH\)独立于通常的集合论公理。（这个结果由耶赫[7]独立得出。）

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a countable dense subset. (One shows first that the countable dense subset is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the "countable dense subset" condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a countable dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2

Iterated Cohen extensions and Souslin's problem*

迭代科恩扩展和苏斯林问题*

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