Homogeneous forcing
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arXiv
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| Format: | Preprint |
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2026
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| author | Shelah, Saharon |
| author_facet | Shelah, Saharon |
| contents | Assume $κ= κ^{< κ}$ (usually $\aleph_0$ or an inaccessible).
We shall deal with iterated forcings preserving ${}^{κ>}{\rm Ord}$ 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 $<κ$ and the $κ^+$-cc, and be strategically $<κ$-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 ${\rm ZF} + {\rm DC}_κ +$ ``modulo this ideal, every set is equivalent to a $κ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_17949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Homogeneous forcing Shelah, Saharon Logic 03E35, 03E25, 03E15 Assume $κ= κ^{< κ}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{κ>}{\rm Ord}$ 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 $<κ$ and the $κ^+$-cc, and be strategically $<κ$-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 ${\rm ZF} + {\rm DC}_κ +$ ``modulo this ideal, every set is equivalent to a $κ$-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. |
| title | Homogeneous forcing |
| topic | Logic 03E35, 03E25, 03E15 |
| url | https://arxiv.org/abs/2603.17949 |