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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2511.01011 |
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| _version_ | 1866912801301200896 |
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| author | Malicki, Maciej |
| author_facet | Malicki, Maciej |
| contents | A co-valuation is, essentially, a minimal finite cover. We introduce a logic based on co-valuations, which play the role of valuations of free variables in classical first-order logic, and show that the fundamental tools of model theory -- such as ultraproducts, compactness, and omitting types -- can be developed in this setup. Using a recently discovered duality between certain countable posets and second-countable compact $T_1$ spaces, we show that these spaces are counterparts of countable universes in first-order logic. Thus, although no topology appears in the initial formulation, the logic of co-valuations turns out to be naturally suited for studying compact topological objects. Standard topological notions, such as connectedness and covering dimension, are easily expressible, and model-theoretic properties, such as atomicity, can be effectively analyzed. The framework also interacts well with Fraïssé-type constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01011 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A logic of co-valuations Malicki, Maciej Logic A co-valuation is, essentially, a minimal finite cover. We introduce a logic based on co-valuations, which play the role of valuations of free variables in classical first-order logic, and show that the fundamental tools of model theory -- such as ultraproducts, compactness, and omitting types -- can be developed in this setup. Using a recently discovered duality between certain countable posets and second-countable compact $T_1$ spaces, we show that these spaces are counterparts of countable universes in first-order logic. Thus, although no topology appears in the initial formulation, the logic of co-valuations turns out to be naturally suited for studying compact topological objects. Standard topological notions, such as connectedness and covering dimension, are easily expressible, and model-theoretic properties, such as atomicity, can be effectively analyzed. The framework also interacts well with Fraïssé-type constructions. |
| title | A logic of co-valuations |
| topic | Logic |
| url | https://arxiv.org/abs/2511.01011 |