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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2308.07510 |
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| _version_ | 1866917792408666112 |
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| author | Moreno, Miguel |
| author_facet | Moreno, Miguel |
| contents | We answer one of the main questions in generalized descriptive set theory, the Friedman-Hyttinen-Kulikov conjecture on the Borel reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel reducibility notions of complexity. For any $κ$ satisfying $κ=λ^+=2^λ$ and $2^{\mathfrak{c}}\leqλ=λ^{ω_1}$, we show that if $T$ is a classifiable theory and $T'$ is a non-classifiable theory, then the isomorphism of models of $T'$ is strictly above the isomorphism of models of $T$ with respect to Borel-reducibility.
We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, $T$, the isomorphism of models of $T$ is either analytic co-analytic, or analytically-complete. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07510 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Shelah's Main Gap and the generalized Borel-reducibility Moreno, Miguel Logic We answer one of the main questions in generalized descriptive set theory, the Friedman-Hyttinen-Kulikov conjecture on the Borel reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel reducibility notions of complexity. For any $κ$ satisfying $κ=λ^+=2^λ$ and $2^{\mathfrak{c}}\leqλ=λ^{ω_1}$, we show that if $T$ is a classifiable theory and $T'$ is a non-classifiable theory, then the isomorphism of models of $T'$ is strictly above the isomorphism of models of $T$ with respect to Borel-reducibility. We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, $T$, the isomorphism of models of $T$ is either analytic co-analytic, or analytically-complete. |
| title | Shelah's Main Gap and the generalized Borel-reducibility |
| topic | Logic |
| url | https://arxiv.org/abs/2308.07510 |