On the Realizability of Prime Conjectures in Heyting Arithmetic
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918278114312192 |
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| author | Rosko, Milan |
| author_facet | Rosko, Milan |
| contents | We show that no total functional can uniformly transform $Π_1$ primality into explicit $Σ_1$ witnesses without violating normalization in $\mathsf{HA}$. The argument proceeds through three complementary translations: a geometric interpretation in which compositeness and primality correspond to local and global packing configurations; a proof-theoretic analysis demonstrating the impossibility of uniform $Σ_1$ extraction; and a recursion-theoretic formulation linking these constraints to the absence of total Skolem functions in $\mathsf{PA}$. The formal analysis in constructive logic is followed by heuristic remarks interpreting the results in informational terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07774 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Realizability of Prime Conjectures in Heyting Arithmetic Rosko, Milan Logic Logic in Computer Science 03F55 (primary), 03F03, 11A41, 68Q05, 11U05 (secondary) F.4.1; F.1.1; F.3.2; G.1.0; G.2.1 We show that no total functional can uniformly transform $Π_1$ primality into explicit $Σ_1$ witnesses without violating normalization in $\mathsf{HA}$. The argument proceeds through three complementary translations: a geometric interpretation in which compositeness and primality correspond to local and global packing configurations; a proof-theoretic analysis demonstrating the impossibility of uniform $Σ_1$ extraction; and a recursion-theoretic formulation linking these constraints to the absence of total Skolem functions in $\mathsf{PA}$. The formal analysis in constructive logic is followed by heuristic remarks interpreting the results in informational terms. |
| title | On the Realizability of Prime Conjectures in Heyting Arithmetic |
| topic | Logic Logic in Computer Science 03F55 (primary), 03F03, 11A41, 68Q05, 11U05 (secondary) F.4.1; F.1.1; F.3.2; G.1.0; G.2.1 |
| url | https://arxiv.org/abs/2511.07774 |