Homogeneous forcing

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.