Matrix iterations with vertical support restrictions

Abstract

We use coherent systems of FS iterations on a power set, which can be seen as matrix iteration that allows restriction on arbitrary subsets of the vertical component, to prove general theorems about preservation of certain type of unbounded families on definable structures and of certain mad families (like those added by Hechler's poset for adding an a.d. family) regardless of the cofinality of their size. In particular, we define a class of posets called $\sigma$-Frechet-linked and show that they work well to preserve mad families, and unbounded families on ${\omega }^{\omega }$. As applications of this method, we show that a large class of FS iterations can preserve the mad family added by Hechler's poset (regardless of the cofinality of its size), and the consistency of a constellation of Cichoń's diagram with 7 values where two of these values are singular.

arXiv commentAccepted in ALC 14 & 15 Proceedings. 28 pages, 6 figures, 1 table

References in corpus (7)

• Decisive creatures and large continuum
• Creature forcing and large continuum: The joy of halving
• Filter-linkedness and its effect on preservation of cardinal characteristics
• Another ordering of the ten cardinal characteristics in Cichoń's diagram
• Template iterations with non-definable ccc forcing notions
• Models of some cardinal invariants with large continuum
• Some models produced by 3D iterations

Cited by in corpus (8)

• Filter-linkedness and its effect on preservation of cardinal characteristics
• On cardinal characteristics of Yorioka ideals
• Controlling cardinal characteristics without adding reals
• Preservation of splitting families and cardinal characteristics of the continuum
• A note on "Another ordering of the ten cardinal characteristics in Cichoń's Diagram" and further remarks
• Uniformity numbers of the null-additive and meager-additive ideals
• More about the cofinality and the covering of the ideal of strong measure zero sets
• Cichoń's maximum with evasion number