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arXiv · 2511.18266

Nonstandard Witnesses and Observational Barriers for {\Pi}0_1 Sentences in ZFC: Standard Cuts, Uniform Reflection Failure, and the Semantic Void

Yusei Fukumoto
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Abstract

We isolate a model-theoretic "standard-cut" phenomenon for true Pi0_1 sentences: if a model M satisfies ZFC + not-phi, then omega^M is not the standard omega, and any internal "witness" to not-phi is computationally inaccessible by Tennenbaum's theorem. Such a witness exists only to maintain syntactic consistency and carries no standard observational semantics. On the proof-theoretic side, we attribute the gap between pointwise verifiability and global provability to a failure of Uniform Reflection. We formalize this as a syntactic self-description failure SDF(T, phi) for proof systems T. Under this failure we obtain an observational barrier: Con(T) implies not Prov_T(phi). In this sense, undecidability in ZFC for Pi0_1 sentences does not describe any observable mathematical reality; it marks a "semantic void", a structural shadow arising not from a standard counterexample but from the expressive limitations of the formal system. We illustrate this with a fixed arithmetical representative of the Riemann Hypothesis.

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Yusei Fukumoto. 2025-11-23. Nonstandard Witnesses and Observational Barriers for {\Pi}0_1 Sentences in ZFC: Standard Cuts, Uniform Reflection Failure, and the Semantic Void. https://arxiv.org/abs/2511.18266

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