Fragility Spectrum: Measuring Resilience in Model-Theoretic Properties under Language Expansions

TL;DR

The paper introduces the fragility spectrum, a new quantitative framework to measure how resilient model-theoretic properties are under language expansions, with applications to classification and complexity analysis.

Contribution

It develops the fragility spectrum concept, axiomatizes fragility operators, and connects these ideas to Shelah's hierarchy and other classification tools in model theory.

Findings

ACF$_0$ exhibits infinite stability fragility

Th($\mathbb{Q}, +$) has fragility 1 for $\omega$-stability

Identifies collapse modes and stratification theorems

Abstract

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 $\omega$-stability). Connections to DOP, ranks, and external definability refine classifications. Extended proofs, applications to other logics, and open problems enhance the discourse.