Preservation of cardinal characteristics in shattered Cohen iterations

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Let b\mathfrak b, d\mathfrak d, and c\mathfrak c denote the bounding number, dominating number, and continuum, respectively. Assume

b=ℵ1,d=c=ν,\mathfrak b=\aleph_1,\qquad \mathfrak d=\mathfrak c=\nu,

and let κ,λ\kappa,\lambda be regular cardinals with ℵ1≤κ<λ≤ν\aleph_1\leq\kappa<\lambda\leq\nu. Consider the extension by the shattered iteration of Cohen forcing. Preservation conjecture. In this extension,

b=add(M)=ℵ1\mathfrak b=\mathsf{add}(\mathcal M)=\aleph_1

and

d=cof(M)=c=ν.\mathfrak d=\mathsf{cof}(\mathcal M)=\mathfrak c=\nu.

The preceding results establish analogous preservation for the null ideal, while the authors state that they believe the corresponding values for the meager ideal and the bounding and dominating numbers are preserved but cannot prove it. The conjecture is open.

References

Primary source

Joerg Brendle, “Shattered Iterations”, arXiv:2302.05069 (2023).

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