Sh:1257

• Shelah, S. Homogeneous forcing, starting again. Preprint.
• Abstract:
  Assume \kappa = \kappa^{< \kappa} (usually \aleph_0 or an inaccessible).

  We shall deal with iterated forcings preserving {}^{\kappa>}Ord and not collapsing cardinals along a linear order. The aim is to have homogeneous ones, so that we have 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 ZF + 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 have homogeneity of \mathbb{Q}, and we do not have automorphisms mapping names of \mathbb{Q}-reals onto each other. However, for some reasonable forcing notions, there are no other \mathbb{Q}-reals! This was the reason for introducing and investigating saccharinity in other works with Jakob Kellner and with Haim Horowitz.

• Version 2025-03-31 (29p)

@article{Sh:1257,
 author = {Shelah, Saharon},
 title = {{Homogeneous forcing, starting again}}
}