Template iterations with non-definable ccc forcing notions

@article{Meja2015TemplateIW,
  title={Template iterations with non-definable ccc forcing notions},
  author={Diego Alejandro Mej{\'i}a},
  journal={Ann. Pure Appl. Log.},
  year={2015},
  volume={166},
  pages={1071-1109}
}
  • D. Mejía
  • Published 21 May 2013
  • Mathematics
  • Ann. Pure Appl. Log.

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References

SHOWING 1-10 OF 33 REFERENCES
Mad families, splitting families and large continuum
TLDR
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 .
TOOLS FOR YOUR FORCING CONSTRUCTION
A preservation theorem is a theorem of the form: "If hP�,Q� : � < �i is an iteration of forcing notions, and every Qsatisfies ' in V P� , then Psatisfies '." We give a simplified version of a general
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
Iterations of Boolean algebras with measure
TLDR
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.
Larger Cardinals in Cichon's Diagram
We prove that in many situations it is consistent with ZFC that part of the invariants involved in Cichon's diagram are equal to κ while the others are equal to λ , where κ λ are both arbitrary
Adjoining Dominating Functions
If dominating functions in ω ω are adjoined repeatedly over a model of GCH via a finite-support c.c.c. iteration, then in the resulting generic extension there are no long towers, every well-ordered
Matrix iterations and Cichon’s diagram
TLDR
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.
INVARIANTS OF MEASURE AND CATEGORY
The purpose of this chapter is to discuss various results concerning the relationship between measure and category. The focus is on set-theoretic properties of the associated ideals, particularly,
Ultrafilters with small generating sets
It is consistent, relative to ZFC, that the minimum number of subsets ofω generating a non-principal ultrafilter is strictly smaller than the dominating number. In fact, these two numbers can be any
Set theory - an introduction to independence proofs
  • K. Kunen
  • Mathematics
    Studies in logic and the foundations of mathematics
  • 1983
TLDR
The Foundations of Set Theory and Infinitary Combinatorics are presented, followed by a discussion of easy Consistency Proofs and Defining Definability.
...
...