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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.05263 |
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| _version_ | 1866917127202537472 |
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| author | Łełyk, Mateusz Walsh, James |
| author_facet | Łełyk, Mateusz Walsh, James |
| contents | There is no recursively enumerable sequence of sufficiently strong 2-consistent r.e. theories such that each proves the $2$-consistency of the next. Montalbán and Shavrukov independently asked whether this result generalizes to $0'$-recursive sequences. We consider a general version of this problem: For arbitrary $n$, for which complexity classes $Γ$ are there $Γ$-definable sequences of $n$-consistent r.e. theories each of which proves the $n$-consistency of the next? The answer to this question depends not only on $n$ and $Γ$ but also on the manner in which sequences are encoded in arithmetic. We provide positive answers for certain encodings and negative answers for others. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05263 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Descending sequences in reflection hierarchies Łełyk, Mateusz Walsh, James Logic There is no recursively enumerable sequence of sufficiently strong 2-consistent r.e. theories such that each proves the $2$-consistency of the next. Montalbán and Shavrukov independently asked whether this result generalizes to $0'$-recursive sequences. We consider a general version of this problem: For arbitrary $n$, for which complexity classes $Γ$ are there $Γ$-definable sequences of $n$-consistent r.e. theories each of which proves the $n$-consistency of the next? The answer to this question depends not only on $n$ and $Γ$ but also on the manner in which sequences are encoded in arithmetic. We provide positive answers for certain encodings and negative answers for others. |
| title | Descending sequences in reflection hierarchies |
| topic | Logic |
| url | https://arxiv.org/abs/2512.05263 |