On Formally Undecidable Propositions of Nondeterministic Complexity and Related Classes

Martin Kolář

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arXiv · 2604.07406

On Formally Undecidable Propositions of Nondeterministic Complexity and Related Classes

Martin Kolář

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Abstract

The definition of \NP\ requires, for each member language~$L$, a polynomial-time checking relation~$R$ and a constant~$k$ such that $w \in L \iff \exists y\,(|y| \leq |w|^k \wedge R(w,y))$. We show that this biconditional instantiates, for each member language, Hilbert's triple: a sound, complete, decidable proof system in which truth-in-$L$ and bounded provability coincide by fiat. We show further that the polynomial-time restriction on~$R$ does not exclude G\"odel's proof-checking relation, which is itself polynomial-time and fits the definition as a literal instance. Hence \NP, taken as a totality over all polynomial-time~$R$, contains languages for which the biconditional asserts a property that G\"odel's First Incompleteness Theorem prohibits. The semantic definition of \NP\ is unsatisfiable, for the same reason that Hilbert's Program is.

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Martin Kolář. 2026-04-08. On Formally Undecidable Propositions of Nondeterministic Complexity and Related Classes. https://arxiv.org/abs/2604.07406

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