Splitting and reaping numbers in shattered Cohen or Hechler iterations

From papers

Let s\mathfrak s and r\mathfrak r denote the splitting and reaping numbers, and let c\mathfrak c denote the continuum. Consider models obtained by the shattered iteration of either Cohen or Hechler forcing. Splitting–reaping conjecture. In these models,

s=1andr=c=μ.\mathfrak s=\aleph_1\qquad\text{and}\qquad \mathfrak r=\mathfrak c=\mu.

For the corresponding shattered Mathias iteration, the theorem computes these characteristics at the iteration parameters, whereas for the Cohen and Hechler models only the bounds sκ\mathfrak s\leq\kappa and rλ\mathfrak r\geq\lambda are known. The conjecture is motivated by preservation of s=1\mathfrak s=\aleph_1 under finite-support iterations of Suslin ccc forcing over models of CH, but remains open.

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Sources & referencesView supporting material

Primary source

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

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