Tukey relations between the null ideal and classical tree forcing ideals
零理想与经典树强迫理想之间的 Tukey 关系
基本信息
- 批准号:276055566
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:德国
- 项目类别:Research Grants
- 财政年份:2015
- 资助国家:德国
- 起止时间:2014-12-31 至 2021-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
A map from an ideal I into another ideal J is called Tukey reduction if every subset of I whose union is not in I gets mapped onto a subset of J with the same property. If I is in this sense Tukey reducible to J, then add(I), its additivity coefficient, is at least as big as add(J). Motivation for the main open problem of this proposal is my latest result which says that the meager ideal is Tukey reducible to the Mycielski ideal. This is the tree ideal associated with Silver forcing. By a famous result of Pawlikowski, the meager ideal is also Tukey reducible to the Lebesgue null ideal. Putting the two together, one would like to know whether my result can be strengthened in the sense that even the null ideal is Tukey reducible to the Mycielski ideal.The Mycielski ideal is one of the classical tree forcing ideals. For most other ones of theses like the Laver, Miller, and the Sacks ideal it has been known for a long time that the meager ideal is not Tukey reducible to any of them. Core element of the proof in all cases has been the construction of an amoeba forcing that does not add Cohen reals. By iterating this it was possible to construct a model in which the additivity of the meager ideal is smaller than the one of the respective tree ideal. If it turns out that we are unable to give a positive answer to our main question, we shall try to construct a Silver amoeba forcing that does not add random reals. Then in a similar way it should be possible to show that consistently the additivity of the null ideal is smaller than the one of the Mycielski ideal. This would imply a negative answer to our problem.Another goal of our project is to prove that the additivities of different such tree ideals are independent. A more modest goal would be to show that there are no provable Tukey reductions between any two of them.Looking at the very definition of these tree ideals suggests that one cannot really understand there behaviour without understanding the antichain structure of the underlying tree forcing. This intuition has been proved to be right by the proof of my latest result mentioned above, where a major ingredient is a result about Silver antichains from my recent work with my student Marek Wyszkowski.I apply that this project be worked out in cooperation with Prof. Saharon Shelah, one of the most prominent logicians of our time. He has done great and deep work in the area of the problems mentioned, such as null ideal, tree forcings and antichain structures. Over many years we have been having a very fruitful cooperation.
如果 I 的并集不在 I 中的每个子集都映射到具有相同属性的 J 子集,则从理想 I 到另一个理想 J 的映射称为 Tukey 约简。如果 I 在这种意义上 Tukey 可简化为 J,则其可加性系数 add(I) 至少与 add(J) 一样大。该提案的主要开放问题的动机是我的最新结果,该结果表明,微薄的理想是 Tukey 可简化为 Mycielski 理想。这是与银强迫相关的理想树。根据 Pawlikowski 的一个著名结果,Tukey 也可以将微薄理想简化为 Lebesgue 零理想。将两者放在一起,人们想知道我的结果是否可以在以下意义上得到加强:即使零理想也可以 Tukey 还原为 Mycielski 理想。Mycielski 理想是经典的树强迫理想之一。对于大多数其他的理论,如拉沃尔、米勒和萨克斯的理想,人们很早就知道,微薄的理想是图基不能简化为其中任何一个的。在所有情况下,证明的核心要素都是构造不添加科恩实数的阿米巴强迫。通过迭代,可以构建一个模型,其中微理想的可加性小于相应树理想的可加性。如果事实证明我们无法对我们的主要问题给出肯定的答案,我们将尝试构建一个不添加随机实数的银阿米巴力。然后以类似的方式应该可以证明零理想的可加性始终小于 Mycielski 理想的可加性。这意味着我们的问题的答案是否定的。我们项目的另一个目标是证明不同的此类树理想的可加性是独立的。一个更温和的目标是表明它们中的任何两个之间都不存在可证明的 Tukey 约简。查看这些树理想的定义表明,如果不了解底层树强迫的反链结构,就无法真正理解其行为。这种直觉已经被我上面提到的最新结果的证明证明是正确的,其中一个主要成分是我最近与我的学生 Marek Wyszkowski 一起工作的关于银反链的结果。我申请与我们这个时代最杰出的逻辑学家之一 Saharon Shelah 教授合作制定这个项目。他在上述问题领域做了伟大而深入的工作,例如零理想、树强迫和反链结构。多年来我们的合作非常富有成效。
项目成果
期刊论文数量(0)
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Professor Dr. Otmar Spinas其他文献
Professor Dr. Otmar Spinas的其他文献
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