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Homogeneous forcing

Saharon Shelah

Abstract

Assume $κ= κ^{< κ}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{κ>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this, which we will focus on, is for the forcings to have support $<κ$ and the $κ^+$-cc, and be strategically $<κ$-complete. The aim is to have homogeneous forcings, so that the iteration has many automorphisms. In addition to the inherent interest, such iterations are helpful for considering some natural ideals on ${}^\kappa2$, in order to get a model of ${\rm ZF} + {\rm DC}_κ +$ ``modulo this ideal, every set is equivalent to a $κ$-Borel one." But here we only have many automorphisms of the index set $L$ and therefore of the iteration of iterands $\mathbb{Q} $; we do not necessarily have homogeneity of $\mathbb{Q} $, and we do not have automorphisms mapping other names of $\mathbb{Q} $-reals onto each other. %\notemgrimes{What are the other names? Where do they come from?} However, for some reasonable forcing notions, there are no other $\mathbb{Q} $-reals! This was the reason for introducing and investigating saccharinity in earlier works with Jakob Kellner and with Haim Horowitz.

Homogeneous forcing

Abstract

Assume (usually or an inaccessible). We shall deal with iterated forcings preserving and not collapsing cardinals along a linear order . A sufficient condition for this, which we will focus on, is for the forcings to have support and the -cc, and be strategically -complete. The aim is to have homogeneous forcings, so that the iteration has many automorphisms. In addition to the inherent interest, such iterations are helpful for considering some natural ideals on , in order to get a model of ``modulo this ideal, every set is equivalent to a -Borel one." But here we only have many automorphisms of the index set and therefore of the iteration of iterands ; we do not necessarily have homogeneity of , and we do not have automorphisms mapping other names of -reals onto each other. %\notemgrimes{What are the other names? Where do they come from?} However, for some reasonable forcing notions, there are no other -reals! This was the reason for introducing and investigating saccharinity in earlier works with Jakob Kellner and with Haim Horowitz.
Paper Structure (9 sections, 1 theorem, 10 equations)

This paper contains 9 sections, 1 theorem, 10 equations.

Table of Contents

  1. Introduction
  2. Aim
  3. Background
  4. Preliminaries
  5. The frame
  6. Investigating $\mathbf{p} \in \mathbf{P}$
  7. From $\mathbf{M}$ and $\mathbf{P}$ to $\mathbf{N}$ and $\mathbf{Q}$
  8. Homogeneity
  9. Variants, and the theorem

Key Result

Theorem 5.3

Adopting the context of i2, we have $(A) \Rightarrow (B)$ and $(A) \wedge (B) \Rightarrow (C)$, where

Theorems & Definitions (62)

  • Definition 2
  • Definition 5
  • Definition 7
  • Remark 8
  • Example 9
  • Claim 10
  • proof : Proof.
  • Definition 1.2
  • Definition 1.3
  • Claim 1.4
  • ...and 52 more