Sets of reals with maximality properties

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540126
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kobe University
• Principal Investigator: BRENDLE Jorg 神戸大学, その他の研究科, 教授 (70301851)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 数学基礎論 / 集合論 / トポロジー / 測度論 / 強制法

Outline of Final Research Achievements

We investigated the structure of the real numbers and of its subsets from the points of view of combinatorial set theory and descriptive set theory. A main focus of this research was to carry out independence proofs, using the method of forcing, about aspects of sets of reals satisfying certain maximality properties, like maximal almost disjoint families, ultrafilters, gaps, and towers.
We obtained new results about cardinal invariants which are defined as the least size of such families with maximality properties, and their relationship with other classical cardinal invariants of the continuum, showing for example that the closed almost disjointness number can consistently be both larger and smaller than the unbounding number.
We also proved new results about the possible complexity of such families in the projective hierarchy, showing for example that the existence of a coanalytic maximal almost disjoint family is consistent with large unbounding number.

Free Research Field

集合論