Cichoń’s diagram and localisation cardinals

@article{Goldstern2021CichosDA,
  title={Cichoń’s diagram and localisation cardinals},
  author={Martin Goldstern and Lukas Daniel Klausner},
  journal={Archive for Mathematical Logic},
  year={2021},
  volume={60},
  pages={343 - 411}
}
We reimplement the creature forcing construction used by Fischer et al. (Arch Math Log 56(7–8):1045–1103, 2017. 10.1007/S00153-017-0553-8. arXiv:1402.0367 [math.LO]) to separate Cichoń’s diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our construction by adding uncountably many additional cardinal characteristics, sometimes referred to as localisation cardinals. 

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References

SHOWING 1-10 OF 25 REFERENCES
There Can Be Many Different Uniformity Numbers of Yorioka Ideals
Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise
Norms on Possibilities I: Forcing With Trees and Creatures
We present a systematic study of the method of "norms on possibilities" of building forcing notions with keeping their properties under full control. This technique allows us to answer several open
Cichoń’s maximum without large cardinals
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
Many simple cardinal invariants
TLDR
It is shown that it is possible to simultaneously force ℵ1 many different values for different functions (f,g) and there may be many distinct uniformII11 characteristics.
Controlling cardinal characteristics without adding reals
We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new [Formula: see text]-sequences (for some regular [Formula: see text]). As an application, we
Another ordering of the ten cardinal characteristics in Cichoń's diagram
It is consistent that א1 < add(N ) < add(M) = b < cov(N ) < non(M) < cov(M) = 2 א0 . Assuming four strongly compact cardinals, it is consistent that א1 < add(N ) < add(M) = b < cov(N ) < non(M) <
Cichoń's maximum
Assuming four strongly compact cardinals, it is consistent that all entries in Cichon's diagram are pairwise different, more specifically that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) <
COHERENT SYSTEMS OF FINITE SUPPORT ITERATIONS
TLDR
A forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets is introduced, which are referred to as 3D-coherent systems.
Simple Cardinal Characteristics of the Continuum
We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (boldface Sigma^0_2
...
1
2
3
...