Title:  Iterated ccc forcing along a linear order and saccharinity






Abstract: Assume $\theta = \aleph_0$ or $\theta = \theta^{<\theta} > \aleph_0$, usually an inaccessible.

We shall deal with iterated forcings preserving ${}^{\theta>}\Ord$ and not collapsing cardinals along a linear order. The aim is to have homogeneous ones, so that for some natural ideals we get a model of $\ZF + \DC_\theta\ +$ ``modulo a natural ideal of ${}^\theta2$, every set is equivalent to a $\theta$-Borel one."

The main application is improving the consistency result of Horowitz-Shelah \cite{Sh:1067}  and  Kellner-Shelah \cite{Sh:859} on saccharinity.The individual iterand is  the one from the work with Horowitz