Tukey relations between the null ideal and classical tree forcing ideals

从理想I到另一个理想J的映射叫做Tukey约简如果I的每个子集其并集不在I中被映射到具有相同性质的J的子集上。如果I在这个意义上可以被Tukey约化为J，那么add(I)，它的可加性系数，至少和add(J)一样大。这个提议的主要公开问题的动机是我最近的结果，它说，贫乏的理想可以近似为米切尔斯基理想。这是与银强迫相关的理想树。根据Pawlikowski的一个著名的结果，微理想也可以被Tukey约化为勒贝格零理想。把这两者放在一起，我们想知道我的结果是否可以被强化到即使是零理想也可以被杜克约化为米捷尔斯基理想。米歇尔斯基理想是经典的树强迫理想之一。对于大多数其他的理论，比如拉弗，米勒和萨克斯的理想，人们早就知道，贫乏的理想并不能完全还原为它们中的任何一个。在所有情况下，证明的核心要素都是构造一个不加科恩实数的变形虫力。通过迭代，有可能构建一个模型，其中贫乏理想的可加性小于各自树理想的可加性。如果结果证明我们不能对我们的主要问题给出一个肯定的答案，我们将尝试构造一个不添加随机实数的银变形虫力。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米捷尔斯基理想的可加性。这意味着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树理想的可加性是独立的。一个更温和的目标应该是表明，这两个国家之间没有任何可证明的土耳其减少。看看这些理想树的定义就会发现，如果不理解潜在的树力的反链结构，就无法真正理解它们的行为。这种直觉已经被我上面提到的最新结果证明是正确的，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。我申请与当代最杰出的逻辑学家之一萨哈拉隆·希拉教授合作制定这个项目。他在上述问题领域做了大量深入的工作，如零理想、树强迫和反链结构。多年来，我们一直有着卓有成效的合作。

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.