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. 

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References

SHOWING 1-10 OF 36 REFERENCES

The left side of Cichoń’s diagram

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

Matrix iterations and Cichon’s diagram

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.

Iterations of Boolean algebras with measure

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

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)

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

  • J. Brendle
  • Mathematics
    Journal 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