SUBCOMPLETE FORCING, TREES, AND GENERIC ABSOLUTENESS
SUBCOMPLETE FORCING, TREES, AND GENERIC ABSOLUTENESS
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次完全强迫、树和一般绝对性
DOI：
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发表时间：
2017
期刊：
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通讯作者：
Kaethe Minden
中科院分区：
文献类型：
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作者：
G. Fuchs;Kaethe Minden
摘要研究了高度为ω1的树的性质及其在次完全强迫下的保持性。我们证明了次完全强迫不能给ω1-树增加一个新的分支。我们引入片段的次完全性，这是保存的次完全强迫，并使用这些，以表明某些强形式的刚性的Suslin树保存的次完全强迫。最后，我们探讨在什么情况下，次完全强迫保持Aronszajn树的高度和宽度ω1。我们表明，这是情况下，如果CH失败，如果CH持有，那么这是情况下，当且仅当有界次完全强制公理持有。最后，我们探讨了有界强迫公理，高度和宽度为ω1的Aronszajn树的保持性和${
m{Sigma }}_1^1$-语句在一阶结构上的大小为ω1，也适用于其他强制的典型类。
Abstract We investigate properties of trees of height ω1 and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an ω1-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcing. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width ω1. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width ω1 and generic absoluteness of ${
m{Sigma }}_1^1$-statements over first order structures of size ω1, also for other canonical classes of forcing.