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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.19216 |
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| _version_ | 1866909863399915520 |
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| author | Gilson, Frank |
| author_facet | Gilson, Frank |
| contents | We determine the ZF-provable modal logic of the modality $\Box_{\mathrm{sym}}$, where $\Box_{\mathrm{sym}}φ$ means '$φ$ holds in every finite symmetry-preserving iteration' of the symmetric method. We prove that the exact logic is S4. Soundness (axioms T and 4) follows from reflexivity and transitivity of the underlying accessibility relation. Exactness is obtained by (i) a non-amalgamation lemma showing that axiom (.2) fails for finite symmetry-preserving iterations (no common finite symmetry-preserving iteration above the parent), and (ii) a $p$-morphism/finite-frame realization producing, within ZF, models whose $\Box_{\mathrm{sym}}$-theory matches any finite reflexive-transitive frame. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19216 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Modal Logic of Finitely Symmetry-Preserving Iterated Extensions is Exactly S4 Gilson, Frank Logic We determine the ZF-provable modal logic of the modality $\Box_{\mathrm{sym}}$, where $\Box_{\mathrm{sym}}φ$ means '$φ$ holds in every finite symmetry-preserving iteration' of the symmetric method. We prove that the exact logic is S4. Soundness (axioms T and 4) follows from reflexivity and transitivity of the underlying accessibility relation. Exactness is obtained by (i) a non-amalgamation lemma showing that axiom (.2) fails for finite symmetry-preserving iterations (no common finite symmetry-preserving iteration above the parent), and (ii) a $p$-morphism/finite-frame realization producing, within ZF, models whose $\Box_{\mathrm{sym}}$-theory matches any finite reflexive-transitive frame. |
| title | The Modal Logic of Finitely Symmetry-Preserving Iterated Extensions is Exactly S4 |
| topic | Logic |
| url | https://arxiv.org/abs/2510.19216 |