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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2411.11432 |
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| _version_ | 1866915128977391616 |
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| author | Šujan, Timotej |
| author_facet | Šujan, Timotej |
| contents | In this paper, we argue that while the concept of a set-theoretic paradox (or paradoxical set) can be relatively well-defined within a formal setting, the concept of a set-theoretic hypodox (or hypodoxical set) remains significantly less clear--especially if the self-membership assertion of the co-Russell set, $\{x:x\in x\}$, is classified as hypodoxical, whereas other set-theoretic sentences with no apparent connection to paradoxes are not. Furthermore, we demonstrate in detail how a contradiction can be derived in Na\"ıve Set Theory by exploiting the unique properties of the co-Russell set, relying on the Fixed Point Theorem of Na\"ıve Set Theory. This result suggests that the boundary between paradoxes and hypodoxes may not be as clear-cut as one might assume. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Set-Theoretic Hypodoxes and co-Russell's Paradox Šujan, Timotej Logic In this paper, we argue that while the concept of a set-theoretic paradox (or paradoxical set) can be relatively well-defined within a formal setting, the concept of a set-theoretic hypodox (or hypodoxical set) remains significantly less clear--especially if the self-membership assertion of the co-Russell set, $\{x:x\in x\}$, is classified as hypodoxical, whereas other set-theoretic sentences with no apparent connection to paradoxes are not. Furthermore, we demonstrate in detail how a contradiction can be derived in Na\"ıve Set Theory by exploiting the unique properties of the co-Russell set, relying on the Fixed Point Theorem of Na\"ıve Set Theory. This result suggests that the boundary between paradoxes and hypodoxes may not be as clear-cut as one might assume. |
| title | Set-Theoretic Hypodoxes and co-Russell's Paradox |
| topic | Logic |
| url | https://arxiv.org/abs/2411.11432 |