Delta12-Sets of Reals
Delta12-Sets of Reals
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Delta12-实数集
DOI：
10.1016/0168-0072(89)90016-x
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发表时间：
1989
期刊：
影响因子：
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通讯作者：
S. Shelah
中科院分区：
文献类型：
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作者：
Jaime I. Ihoda;S. Shelah
我们考虑并给出如下形式的一个完全解：（∗）每个X - 1实数集具有p1的性质意味着每个X - 2实数集具有p2的性质，对于X - 1， X - 2 λ {Δ 1 2， Π 1 1, σ 1 2， Π 1 2}，并且P - 1， P - 2属于“拉姆齐”，“K σ-正则”，当然还有“勒贝格可测”和“贝尔范畴”。自然地，我们被引导去寻找这些性质的特征（通过强迫）。毫不奇怪，除了一些琐碎的暗示，我们得到了许多一致性结果，但“幸运的是”我们得到了相当多的定理（=在ZFC中证明的暗示），特别是在“to be Ramsay”和“K σ-regular”中。1. 定理1。下面是等价的：(a)每个Σ 1 2-set real都是Ramsey。(b)每个Δ 2-set real都是Ramsey。(c)对于每一个r λ r，存在s λ [ω] ω， s为P (D s [r])-泛型/ L [r][D s]（定义见第0节）。对于这一定理，我们提出了一个强迫P (D)（D是一个超滤器在ω上），使超滤器真正“穿过”。1. 定理2。以下是等价的：(a)每一个Σ - 1实数集是K Σ -正则的。(b)每一个Δ - 1实数集都是K σ-正则的。(c)每一个Π - 11实数集都是K σ-正则的。(d)对于每一个r御R，存在f御ω ω， f是ω ω∩L [R]的σ绑定。
We consider and give a complete solution to, implications of the form:(∗) Every X 1-set of reals has the property P 1 implies every X 2-set of reals has the property P 2, for X 1, X 2 ϵ {Δ 1 2, Π 1 1, σ 1 2, Π 1 2}, and where P 1, P 2 are among ‘to be Ramsey’,‘K σ-regular’and of course ‘Lebesgue measurable’and ‘Baire categoricity’. Naturally we are led to look for characterizations of such properties (by forcing). Not surprisingly, excepting the trivial implications, we get many consistency results, but ‘fortunately’we get quite a number of theorems (= implications proved in ZFC), notably among the ‘to be Ramsay’and ‘K σ-regular’. 1. Theorem 1. The following are equivalent:(a) Every Σ 1 2-set of reals is Ramsey.(b) Every Δ 1 2-set of reals is Ramsey.(c) For every r ϵ R there exists s ϵ [ω] ω, s is P (D s [r])-generic over L [r][D s](Definitions are given in Section 0.) For this theorem we develop a forcing P (D)(D an ultrafilter on ω) shooting a real ‘through’the ultrafilter. 1. Theorem 2. The following are equivalent:(a) Every Σ 1 2-set of reals is K σ-regular.(b) Every Δ 1 2-set of reals is K σ-regular.(c) Every Π 1 1-set of reals is K σ-regular.(d) For every rϵ R, there exists fϵ ω ω, f is a σ-bound to ω ω∩ L [r].