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Preservation of splitting families and cardinal characteristics of the continuum

Martin Goldstern, Jakob Kellner, Diego A. Mejía, Saharon Shelah

Abstract

We show how to construct, via forcing, splitting families than are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń's diagram, $\mathfrak{m}(2\text{-Knaster})$, $\mathfrak{p}$, $\mathfrak{h}$, the splitting number $\mathfrak{s}$ and the reaping number $\mathfrak{r}$.

Preservation of splitting families and cardinal characteristics of the continuum

Abstract

We show how to construct, via forcing, splitting families than are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń's diagram, , , , the splitting number and the reaping number .

Paper Structure

This paper contains 17 sections, 27 theorems, 38 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Suitable 2-graphs
  3. Cardinal Characteristics, COB and LCU
  4. Preserving splitting families with symmetric iterations
  5. The single forcing GB
  6. Suitable iterations, nice names and automorphisms
  7. A digression: Self-indexed products
  8. Symmetric small history iterations preserve splitting families
  9. Suslin-lambda-small iterations
  10. The forcing construction for the left hand side
  11. Preliminaries.
  12. The basic forcing construction.
  13. Dealing with b
  14. The other constellations for the Knaster numbers
  15. The alternative order of the left side
  16. ...and 2 more sections

Key Result

Theorem 1.3

Under $\mathrm{GCH}$, for any $k\in[1,\omega)$, there is a cofinality preserving poset $\mathbb{P}_{k}$ forcing that An analogous result holds for the alternative order of Figure fig:cichonorders(b).

Figures (6)

  • Figure 1: Cichoń's diagram (left). In the version on the right, the two "dependent" values ${\mathop{\mathrm{add}}\nolimits(\mathcal{M})}=\min\{\mathfrak b, {\mathop{\mathrm{cov}}\nolimits(\mathcal{M})}\}$ and ${\mathop{\mathrm{cof}}\nolimits(\mathcal{M})}=\max\{{\mathop{\mathrm{non}}\nolimits(\mathcal{M})},\mathfrak d\}$ are removed; the "independent" ones remain (nine entries excluding $\aleph_1$, or ten including it). An arrow $\mathfrak x\rightarrow \mathfrak y$ means that ZFC proves $\mathfrak x\le \mathfrak y$.
  • Figure 2: Cichoń's diagram and the Blass diagram combined. An arrow $\mathfrak x\rightarrow \mathfrak y$ means that ZFC proves $\mathfrak x\le \mathfrak y$.
  • Figure 3: The two known consistent orders where all the (non-dependent) values in Cichoń's diagram are pairwise different. (A) corresponds to the model in GKS, and (B) to the model in KeShTa:1131 (both proven consistent in GKMS2 without large cardinals). Each arrow can be $<$ or $=$ as desired.
  • Figure 4: A finite $2$-graph which cannot be respected by any coloring.
  • Figure 5: A suitable iteration. $\pi_1=\omega_1\pi_0$ is partitioned into $\pi_0$-many intervals of length $\omega_1$, and $B_\delta:=[\omega_1\delta,\omega(\delta+1))$, the set of vertices of the graph $\mathbf{B}_\delta$, is the $\delta$-th interval of this partition. A suitable iteration is a FS product of the $\mathbb{G}_{\mathbf{B}_\delta}$ for $\delta<\pi_0$, followed by a FS iteration of ccc posets. The iterands of the FS iteration that follow are indexed by $\alpha\in[\pi_1,\pi)$.
  • ...and 1 more figures

Theorems & Definitions (79)

  • Definition 1.1
  • Theorem 1.3: GKMS1
  • Definition 2.1
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Remark 2.7
  • ...and 69 more