Rich doctrines and Henkin's Theorem
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914871716610048 |
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| author | Guffanti, Francesca |
| author_facet | Guffanti, Francesca |
| contents | We find a possible interpretation of Henkin's Theorem in the language of existential implicational doctrines. Under some smallness assumption, starting from an implicational existential doctrine, with non-trivial fibers, we construct a new doctrine which is rich -- meaning that for every formula $φ(x)$ there is a constant $c$ such that $\exists xφ(x)$ has the same truth-value of $φ(c)$ -- and consistent. To obtain this result, we add a suitable amount of constants and axioms to the starting doctrine. We then show that a rich consistent doctrine admits an appropriate morphism towards the doctrine of subsets -- a model. Henkin's Theorem for doctrines follows from these two results, modeling our proof on the main lines of the original theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08374 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rich doctrines and Henkin's Theorem Guffanti, Francesca Category Theory Logic We find a possible interpretation of Henkin's Theorem in the language of existential implicational doctrines. Under some smallness assumption, starting from an implicational existential doctrine, with non-trivial fibers, we construct a new doctrine which is rich -- meaning that for every formula $φ(x)$ there is a constant $c$ such that $\exists xφ(x)$ has the same truth-value of $φ(c)$ -- and consistent. To obtain this result, we add a suitable amount of constants and axioms to the starting doctrine. We then show that a rich consistent doctrine admits an appropriate morphism towards the doctrine of subsets -- a model. Henkin's Theorem for doctrines follows from these two results, modeling our proof on the main lines of the original theorem. |
| title | Rich doctrines and Henkin's Theorem |
| topic | Category Theory Logic |
| url | https://arxiv.org/abs/2310.08374 |