# Cichoń's maximum

@article{Goldstern2019CichosM,
title={Cichoń's maximum},
author={Martin Goldstern and Jakob Kellner and Saharon Shelah},
journal={Annals of Mathematics},
year={2019}
}
• Published 8 August 2017
• Materials Science, Mathematics
• Annals of Mathematics
Assuming four strongly compact cardinals, it is consistent that all entries in Cichon's diagram are pairwise different, more specifically that $\aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.$

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## References

SHOWING 1-10 OF 37 REFERENCES
Another ordering of the ten cardinal characteristics in Cichoń's diagram
• Materials Science, Mathematics
Commentationes Mathematicae Universitatis Carolinae
• 2019
It is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) < \mathrm{add}(\mathrm{Meager})= \mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) <
Creature forcing and five cardinal characteristics in Cichoń’s diagram
• Mathematics, Materials Science
Arch. Math. Log.
• 2017
A (countable support) creature construction is used to show that consistently aleph _1=cov_1= {{\mathrm{cov}}}(\mathcal N)< non(N)<non(M)<cof(N), and aleph_0=2ℵ0.
The left side of Cichoń’s diagram
• Mathematics
• 2016
Using a finite support iteration of ccc forcings, we construct a model of
COMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM
• Mathematics
The Journal of Symbolic Logic
• 2018
Under the same assumption, it is consistent that aleph _1 < add( {\cal N) < cov( {cal N}) < non\left ( {\cal M}) > cov( ¬N) < cof( −1) < 2 ( −2) .
Controlling classical cardinal characteristics while collapsing cardinals
• Mathematics
Colloquium Mathematicum
• 2022
Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $\lambda>\kappa$ to
A characterization of the least cardinal for which the Baire category theorem fails
Let k be the least cardinal such that the real Une can be covered by k many nowhere dense sets. We show that k can be characterized as the least cardinal such that "infinitely equal" reals fail to
Controlling cardinal characteristics without adding reals
• Mathematics
J. Math. Log.
• 2021
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
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.