Set-theoretic mereology
We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable,...
| Main Authors: | , |
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| Format: | Journal article |
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Uniwersytet Mikołaja Kopernika w Toruniu
2016
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| _version_ | 1797063926711582720 |
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| author | Hamkins, J Kikuchi, M |
| author_facet | Hamkins, J Kikuchi, M |
| author_sort | Hamkins, J |
| collection | OXFORD |
| description | We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust. |
| first_indexed | 2024-03-06T21:06:58Z |
| format | Journal article |
| id | oxford-uuid:3cccc2ed-09da-40d5-acf6-b7e2aa6284f5 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T21:06:58Z |
| publishDate | 2016 |
| publisher | Uniwersytet Mikołaja Kopernika w Toruniu |
| record_format | dspace |
| spelling | oxford-uuid:3cccc2ed-09da-40d5-acf6-b7e2aa6284f52022-03-26T14:15:41ZSet-theoretic mereologyJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3cccc2ed-09da-40d5-acf6-b7e2aa6284f5Symplectic Elements at OxfordUniwersytet Mikołaja Kopernika w Toruniu2016Hamkins, JKikuchi, MWe consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust. |
| spellingShingle | Hamkins, J Kikuchi, M Set-theoretic mereology |
| title | Set-theoretic mereology |
| title_full | Set-theoretic mereology |
| title_fullStr | Set-theoretic mereology |
| title_full_unstemmed | Set-theoretic mereology |
| title_short | Set-theoretic mereology |
| title_sort | set theoretic mereology |
| work_keys_str_mv | AT hamkinsj settheoreticmereology AT kikuchim settheoreticmereology |