The Axiom of Double Complement and its opposites
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913176328601600 |
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| author | Jeon, Hanul Hanson, James E. |
| author_facet | Jeon, Hanul Hanson, James E. |
| contents | Powell introduced the Axiom of Double Complement ($\mathsf{DCom}$) to give his double-negation interpretation of $\mathsf{ZF}$ into $\mathsf{IZF_{Rep}}$. However, the consistency, strength, and compatibility of $\mathsf{DCom}$ remain open problems. This article aims to survey the compatibility and consistency strength of $\mathsf{DCom}$, its consequence and opposites, which will be named $\mathsf{NDCom}$ and $\mathsf{ADCom}$. We will also develop Lubarsky's Kripke models over $\mathsf{CZF}$ to derive these results.
We will show that $\mathsf{DCom}$ proves the Powerset axiom over $\mathsf{CZF}$ and is independent of $\mathsf{IZF}$. We will also show that $\mathsf{ADCom}$ does not add consistency strength over $\mathsf{CZF}$, by modifying the construction of Lubarsky's model for $\mathsf{CZF+\lnot Pow}$. We will also show that $\mathsf{DCom}$, $\mathsf{ADCom}$, and $\mathsf{NDCom}$ are persistent under realizability under modest conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00861 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Axiom of Double Complement and its opposites Jeon, Hanul Hanson, James E. Logic 03E70, 03F65, 03E35, 03F25 Powell introduced the Axiom of Double Complement ($\mathsf{DCom}$) to give his double-negation interpretation of $\mathsf{ZF}$ into $\mathsf{IZF_{Rep}}$. However, the consistency, strength, and compatibility of $\mathsf{DCom}$ remain open problems. This article aims to survey the compatibility and consistency strength of $\mathsf{DCom}$, its consequence and opposites, which will be named $\mathsf{NDCom}$ and $\mathsf{ADCom}$. We will also develop Lubarsky's Kripke models over $\mathsf{CZF}$ to derive these results. We will show that $\mathsf{DCom}$ proves the Powerset axiom over $\mathsf{CZF}$ and is independent of $\mathsf{IZF}$. We will also show that $\mathsf{ADCom}$ does not add consistency strength over $\mathsf{CZF}$, by modifying the construction of Lubarsky's model for $\mathsf{CZF+\lnot Pow}$. We will also show that $\mathsf{DCom}$, $\mathsf{ADCom}$, and $\mathsf{NDCom}$ are persistent under realizability under modest conditions. |
| title | The Axiom of Double Complement and its opposites |
| topic | Logic 03E70, 03F65, 03E35, 03F25 |
| url | https://arxiv.org/abs/2606.00861 |