Fragility Spectrum: Measuring Resilience in Model-Theoretic Properties under Language Expansions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912538999914496 |
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| author | Adilkhan, Nuraly |
| author_facet | Adilkhan, Nuraly |
| contents | We introduce the fragility spectrum, a quantitative framework to measure the resilience of model-theoretic properties (e.g., stability, NIP, NTP$_2$, decidability) under language expansions. The core is the fragility index $\operatorname{frag}(T, P \to Q)$, quantifying the minimal expansion needed to degrade from property $P$ to $Q$. We axiomatize fragility operators, prove stratification theorems, identify computational, geometric, and combinatorial collapse modes, and position it within Shelah's hierarchy. Examples include ACF$_0$ (infinite fragility for stability) and $\operatorname{Th}(\mathbb{Q}, +)$ (fragility 1 for $ω$-stability). Connections to DOP, ranks, and external definability refine classifications. Extended proofs, applications to other logics, and open problems enhance the discourse. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fragility Spectrum: Measuring Resilience in Model-Theoretic Properties under Language Expansions Adilkhan, Nuraly Logic We introduce the fragility spectrum, a quantitative framework to measure the resilience of model-theoretic properties (e.g., stability, NIP, NTP$_2$, decidability) under language expansions. The core is the fragility index $\operatorname{frag}(T, P \to Q)$, quantifying the minimal expansion needed to degrade from property $P$ to $Q$. We axiomatize fragility operators, prove stratification theorems, identify computational, geometric, and combinatorial collapse modes, and position it within Shelah's hierarchy. Examples include ACF$_0$ (infinite fragility for stability) and $\operatorname{Th}(\mathbb{Q}, +)$ (fragility 1 for $ω$-stability). Connections to DOP, ranks, and external definability refine classifications. Extended proofs, applications to other logics, and open problems enhance the discourse. |
| title | Fragility Spectrum: Measuring Resilience in Model-Theoretic Properties under Language Expansions |
| topic | Logic |
| url | https://arxiv.org/abs/2508.10073 |