Sh:1257

• Shelah, S. Homogeneous forcing. Preprint. arXiv: 2603.17949
• 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. 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.

• Version 2026-03-12 (32p)

@article{Sh:1257,
 author = {Shelah, Saharon},
 title = {{Homogeneous forcing}},
 note = {\href{https://arxiv.org/abs/2603.17949}{arXiv: 2603.17949}},
 arxiv_number = {2603.17949}
}