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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.27077 |
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| _version_ | 1866912986523762688 |
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| author | Simmons, David |
| author_facet | Simmons, David |
| contents | We introduce a formal language GDST (gradualist descriptionalist set theory) with a family of interpretations indexed by ordinals, as well as a sublanguage NMID (the language of not necessarily monotonic inductive definitions), and show that the assertion that all propositions in NMID have well-defined truth values is equivalent to the existence for each $k \in \mathbb N$ of a sequence of ordinals $η_0 < . . . < η_k$ such that for each $i < k$, $η_i$ is $η_{i+1}$-reflecting, a notion we introduce which implies being $Π_n$-reflecting for all $n \in \mathbb N$ (and in particular being admissible and recursively Mahlo). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27077 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gradualist descriptionalist set theory Simmons, David Logic We introduce a formal language GDST (gradualist descriptionalist set theory) with a family of interpretations indexed by ordinals, as well as a sublanguage NMID (the language of not necessarily monotonic inductive definitions), and show that the assertion that all propositions in NMID have well-defined truth values is equivalent to the existence for each $k \in \mathbb N$ of a sequence of ordinals $η_0 < . . . < η_k$ such that for each $i < k$, $η_i$ is $η_{i+1}$-reflecting, a notion we introduce which implies being $Π_n$-reflecting for all $n \in \mathbb N$ (and in particular being admissible and recursively Mahlo). |
| title | Gradualist descriptionalist set theory |
| topic | Logic |
| url | https://arxiv.org/abs/2603.27077 |