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.
Categories: math.LO