# 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.

## 15 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…

### 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…

### GOOD PROJECTIVE WITNESSES

- Mathematics
- 2019

Developing a new forcing notion for adjoining self-coding co nitary permutations, we show that consistently there is a Π2-de nable maximal co nitary group of cardinality μ, where א1 < μ < c. Here Π2…

### 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

- MathematicsArch. Math. Log.
- 2021

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.

### Filter-linkedness and its effect on preservation of cardinal characteristics

- MathematicsAnn. Pure Appl. Log.
- 2021

## References

SHOWING 1-10 OF 36 REFERENCES

### The left side of Cichoń’s diagram

- Mathematics
- 2016

Using a finite support iteration of ccc forcings, we construct a model of…

### Matrix iterations and Cichon’s diagram

- MathematicsArch. Math. Log.
- 2013

Using matrix iterations of ccc posets, it is proved that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon’s diagram.

### Cardinal characteristics, projective wellorders and large continuum

- EconomicsAnn. Pure Appl. Log.
- 2013

### Iterations of Boolean algebras with measure

- MathematicsArch. Math. Log.
- 1989

It is shown that M is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model, and a simple characterization of Martin's Axiom is deduced.

### Mad families, splitting families and large continuum

- MathematicsThe Journal of Symbolic Logic
- 2011

Using a finite support iteration of ccc posets, if μ is a measurable cardinal and μ < κ < λ, then using similar techniques the authors obtain the consistency of .

### Models of some cardinal invariants with large continuum (Forcing extensions and large cardinals)

- Mathematics
- 2013

We extend the applications of the techniques used in Arch Math Logic 52:261-278, 2013, to present various examples of consistency results where some cardinal invariants of the continuum take…

### Larger cardinals in Cichoń's diagram

- MathematicsJournal of Symbolic Logic
- 1991

Abstract We prove that in many situations it is consistent with ZFC that part of the invariants involved in Cichoń's diagram are equal to κ while the others are equal to λ, where κ < λ are both…