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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2503.19833 |
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| _version_ | 1866909574154420224 |
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| author | Wiesnet, Franziskus |
| author_facet | Wiesnet, Franziskus |
| contents | This article presents the concept of material interpretation as a method to transform classical proofs into constructive ones. Using the case study of maximal ideals in $\mathbb{Z}[X]$, it demonstrates how a classical implication $A \to B$ can be rephrased as a constructive disjunction $\neg A \vee B$, with $\neg A$ representing a strong form of negation. The approach is based on on Gödel's Dialectica interpretation, the strong negation, and potentially Herbrand disjunctions. The classical proof that every maximal ideal in $\mathbb{Z}[X]$ contains a prime number is revisited, highlighting its reliance on non-constructive principles such as the law of excluded middle. A constructive proof is then developed, replacing abstract constructs with explicit case distinctions and direct computations in $\mathbb{Z}[X]$. This proof clarifies the logical structure and reveals computational content. The article discusses broader applications, such as Zariski's Lemma, Hilbert's Nullstellensatz, and the Universal Krull-Lindenbaum Lemma, with an emphasis on practical implementation using tools such as Python and proof assistants. The material interpretation offers a promising framework for bridging classical and constructive mathematics, enabling algorithmic implementations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Material Interpretation and Constructive Analysis of Maximal Ideals in $\mathbb{Z}[X]$ Wiesnet, Franziskus Logic Commutative Algebra F.4.1 This article presents the concept of material interpretation as a method to transform classical proofs into constructive ones. Using the case study of maximal ideals in $\mathbb{Z}[X]$, it demonstrates how a classical implication $A \to B$ can be rephrased as a constructive disjunction $\neg A \vee B$, with $\neg A$ representing a strong form of negation. The approach is based on on Gödel's Dialectica interpretation, the strong negation, and potentially Herbrand disjunctions. The classical proof that every maximal ideal in $\mathbb{Z}[X]$ contains a prime number is revisited, highlighting its reliance on non-constructive principles such as the law of excluded middle. A constructive proof is then developed, replacing abstract constructs with explicit case distinctions and direct computations in $\mathbb{Z}[X]$. This proof clarifies the logical structure and reveals computational content. The article discusses broader applications, such as Zariski's Lemma, Hilbert's Nullstellensatz, and the Universal Krull-Lindenbaum Lemma, with an emphasis on practical implementation using tools such as Python and proof assistants. The material interpretation offers a promising framework for bridging classical and constructive mathematics, enabling algorithmic implementations. |
| title | Material Interpretation and Constructive Analysis of Maximal Ideals in $\mathbb{Z}[X]$ |
| topic | Logic Commutative Algebra F.4.1 |
| url | https://arxiv.org/abs/2503.19833 |