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| Format: | Preprint |
| Published: |
2018
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1810.00493 |
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| _version_ | 1866911953660674048 |
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| author | Ulrich, Danielle |
| author_facet | Ulrich, Danielle |
| contents | We introduce a new invariant of Borel reducibility, namely the notion of thickness; this associates to every sentence $Φ$ of $\mathcal{L}_{ω_1 ω}$ and to every cardinal $λ$, the thickness $τ(Φ, λ)$ of $Φ$ at $λ$. As applications, we show that all the Friedman-Stanley jumps of torsion abelian groups are non-Borel complete. We also show that under the existence of large cardinals, if $Φ$ is a sentence of $\mathcal{L}_{ω_1 ω}$ with the Schröder-Bernstein property (that is, whenever two countable models of $Φ$ are biembeddable, then they are isomorphic), then $Φ$ is not Borel complete. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_00493 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Borel Complexity and the Schröder-Bernstein Property Ulrich, Danielle Logic 03C55 We introduce a new invariant of Borel reducibility, namely the notion of thickness; this associates to every sentence $Φ$ of $\mathcal{L}_{ω_1 ω}$ and to every cardinal $λ$, the thickness $τ(Φ, λ)$ of $Φ$ at $λ$. As applications, we show that all the Friedman-Stanley jumps of torsion abelian groups are non-Borel complete. We also show that under the existence of large cardinals, if $Φ$ is a sentence of $\mathcal{L}_{ω_1 ω}$ with the Schröder-Bernstein property (that is, whenever two countable models of $Φ$ are biembeddable, then they are isomorphic), then $Φ$ is not Borel complete. |
| title | Borel Complexity and the Schröder-Bernstein Property |
| topic | Logic 03C55 |
| url | https://arxiv.org/abs/1810.00493 |