Lattice initial segments of the hyperdegrees
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2014
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| _version_ | 1866913580387926016 |
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| author | Shore, Richard A. Kjos-Hanssen, Bjørn |
| author_facet | Shore, Richard A. Kjos-Hanssen, Bjørn |
| contents | We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $% \mathcal{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $ω_{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $ω_{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1408_3147 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Lattice initial segments of the hyperdegrees Shore, Richard A. Kjos-Hanssen, Bjørn Logic 03D We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $% \mathcal{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $ω_{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $ω_{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$. |
| title | Lattice initial segments of the hyperdegrees |
| topic | Logic 03D |
| url | https://arxiv.org/abs/1408.3147 |