The Solver's Paradox in Formal Problem Spaces

TL;DR

This paper explores the logical complexity of decision problems in formal arithmetic, revealing that their difficulty arises from structural impredicativity and reflection principles, not computational limitations.

Contribution

It introduces a new perspective on the logical foundations of complexity theory, emphasizing the role of impredicative structures in problem difficulty.

Findings

Arithmetization induces fixed points requiring reflection beyond finitary methods

Structural impredicativity explains the difficulty of P vs. NP and similar problems

Clarifies the logical status of arithmetized complexity assertions

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.