Homogeneous forcing

TL;DR

This paper explores conditions under which iterated forcing notions preserve cardinals and have many automorphisms, focusing on homogeneous forcings to analyze ideals on ${}^oldsymbol{ ext{kappa}2}$ and models of set theory.

Contribution

It introduces a framework for homogeneous forcing iterations with support $<oldsymbol{ ext{kappa}}$, preserving cardinals and automorphisms, and investigates their applications to ideals and models of set theory.

Findings

Supports preserve cardinals and have many automorphisms.

Homogeneous forcings can be constructed with specific support and chain conditions.

Applications to models of ZF + DC$_oldsymbol{ ext{kappa}}$ with ideal-related properties.

Abstract

Assume $\kappa = \kappa^{< \kappa}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{\kappa>}{\rm Ord}$ 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 $<\kappa$ and the $\kappa^+$-cc, and be strategically $<{\kappa}$-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 ${\rm ZF} + {\rm DC}_\kappa\ +$ ``modulo this ideal, every set is equivalent to a $\kappa$-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.