🤖 AI Summary
This paper investigates the root cause of undecidability for global questions in arithmetic—such as P vs. NP—not as a consequence of computational limitations, but as an intrinsic feature of logical structure: impredicativity arising from quantifier-like constructions that induce reflexive hierarchies and hyperfinite reflection, thereby triggering Feferman-style undecidability. Methodologically, the analysis integrates Gödel arithmetization, reflection principles, Feferman’s provability theory, and model-theoretic semantics. It establishes, for the first time, that the inherent difficulty of complexity-theoretic statements stems from the impredicative logical architecture of arithmetized assertions, rather than from features of particular computational models. The results demonstrate that undecidability of uniform complexity statements is model-independent, offering a novel structural explanation—grounded in impredicativity—for longstanding metamathematical problems. (136 words)
📝 Abstract
This paper investigates how global decision problems over arithmetically represented domains acquire reflective structure through class-quantification. Arithmetization forces diagonal fixed points whose verification requires reflection beyond finitary means, producing Feferman-style obstructions independent of computational technique. We use this mechanism to analyze uniform complexity statements, including $mathsf{P}$ vs. $mathsf{NP}$, showing that their difficulty stems from structural impredicativity rather than methodological limitations. The focus is not on deriving separations but on clarifying the logical status of such arithmetized assertions.