# COHERENT SYSTEMS OF FINITE SUPPORT ITERATIONS

@article{Fischer2018COHERENTSO, title={COHERENT SYSTEMS OF FINITE SUPPORT ITERATIONS}, author={Vera Fischer and Sy-David Friedman and Diego Alejandro Mej{\'i}a and Diana Carolina Montoya}, journal={The Journal of Symbolic Logic}, year={2018}, volume={83}, pages={208 - 236} }

Abstract We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń’s diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a ${\rm{\Delta }}_3^1$ well-order of the reals.

## 16 Citations

### Matrix Iterations with Vertical Support Restrictions

- MathematicsProceedings of the 14th and 15th Asian Logic Conferences
- 2019

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…

### Good projective witnesses

- Mathematics
- 2019

. We develop a new forcing notion for adjoining self-coding coﬁnitary permutations and use it to show that consistently, the minimal cardinality a g of a maximal coﬁnitary group (MCG) is strictly…

### PARTITION FORCING AND INDEPENDENT FAMILIES

- MathematicsThe Journal of Symbolic Logic
- 2022

. We show that Miller partition forcing preserves selective independent families and P -points, which implies the consistency of cof ( N ) = a = u = i < a T = ω 2 . In addition, we show that Shelah’s…

### Some models produced by

- Materials Science
- 2018

We use the techniques in [BFII, \mathrm{M}\mathrm{e}\mathrm{j}\mathrm{l}3\mathrm{b} , FFMM] to construct models, by three‐ dimensional arrays of ccc posets, where many classical cardinal…

### DEFINABLE MINIMAL COLLAPSE FUNCTIONS AT ARBITRARY PROJECTIVE LEVELS

- MathematicsThe Journal of Symbolic Logic
- 2019

A generic extension of Uri Abraham’s minimal $\Delta _3^1$ collapse function is defined by a real a, in which, for a given $n \ge 3$ is a lightface $\Pi _n^1 $ singleton, a effectively codes a cofinal map $\omega \to \omega _1^L $ minimal over L.

### Cichoń’s diagram and localisation cardinals

- MathematicsArchive for Mathematical Logic
- 2020

This work reimplements the creature forcing construction used by Fischer et al. to separate Cichoń’s diagram into five cardinals as a countable support product, and adds uncountably many additional cardinal characteristics, sometimes referred to as localisation cardinals.

### On the bounding, splitting, and distributivity numbers

- Mathematics
- 2022

The cardinal invariants h, b, s of Ppωq are known to satisfy that ω1 ď h ď mintb, su. We prove that all inequalities can be strict. We also introduce a new upper bound for h and show that it can be…

### Cichoń’s maximum without large cardinals

- MathematicsJournal of the European Mathematical Society
- 2021

Cichon's diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and $\sigma$-compact subsets…

### On cardinal characteristics of Yorioka ideals

- MathematicsMath. Log. Q.
- 2019

It is shown that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with the addition and co finality of the ideal of Lebesgue measure zero subsets of the real line.

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