Perfect set dichotomy theorem in generalized Solovay model
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908690494259200 |
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| author | Sakai, Hiroshi Tanno, Toshimasa |
| author_facet | Sakai, Hiroshi Tanno, Toshimasa |
| contents | We prove that the perfect set dichotomy theorem holds in the Solovay model $V ((ω^ω)^{V[G]})$. Namely, for every equivalence relation $E$ on $\mathbb{R}$, either $\mathbb{R}/E$ is well-orderable or there exists a perfect set consisting of $E$-inequivalent reals.
Furthermore we consider a generalization of the Solovay model for an uncountable regular cardinal $μ$ and show the perfect set dichotomy theorem for $μ^μ$ also holds in that model.
We establish the three element basis theorem for uncountable linear orders in the Solovay model for a weakly compact cardinal, in a general form covering the uncountable case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03386 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perfect set dichotomy theorem in generalized Solovay model Sakai, Hiroshi Tanno, Toshimasa Logic We prove that the perfect set dichotomy theorem holds in the Solovay model $V ((ω^ω)^{V[G]})$. Namely, for every equivalence relation $E$ on $\mathbb{R}$, either $\mathbb{R}/E$ is well-orderable or there exists a perfect set consisting of $E$-inequivalent reals. Furthermore we consider a generalization of the Solovay model for an uncountable regular cardinal $μ$ and show the perfect set dichotomy theorem for $μ^μ$ also holds in that model. We establish the three element basis theorem for uncountable linear orders in the Solovay model for a weakly compact cardinal, in a general form covering the uncountable case. |
| title | Perfect set dichotomy theorem in generalized Solovay model |
| topic | Logic |
| url | https://arxiv.org/abs/2512.03386 |