Nonstandard Witnesses and Observational Barriers for Π0_1 Sentences in ZFC: Standard Cuts, Uniform Reflection Failure, and the Semantic Void
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909919512363008 |
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| author | Fukumoto, Yusei |
| author_facet | Fukumoto, Yusei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonstandard Witnesses and Observational Barriers for Π0_1 Sentences in ZFC: Standard Cuts, Uniform Reflection Failure, and the Semantic Void Fukumoto, Yusei Logic 03C62, 03F30, 03D35, 03F40, 11M26 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. |
| title | Nonstandard Witnesses and Observational Barriers for Π0_1 Sentences in ZFC: Standard Cuts, Uniform Reflection Failure, and the Semantic Void |
| topic | Logic 03C62, 03F30, 03D35, 03F40, 11M26 |
| url | https://arxiv.org/abs/2511.18266 |