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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.14400 |
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| _version_ | 1866914019856613376 |
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| author | Simón, Nicolás Sevilla |
| author_facet | Simón, Nicolás Sevilla |
| contents | In this paper, we present a proof of the consistency of the New Foundations set theory ($\mathit{NF}$). $\mathit{NF}$'s main idea is to permit very large sets (including the Universal Set) by restricting set formation to stratified formulas, thereby avoiding the classic set-theoretic paradoxes. Our proof employs a new forcing method incorporating concepts from fuzzy logic. A brief outline of the proof can be as follows: (1) We extend $ZF$ to Fuzzy $\mathit{ZF}$ with a membership function $μ$ over $D=\mathbb{Q} \cap [0,1]$; (2) we define Fuzzy $\mathit{NF}$ as $Σ$, and (3) we derive a crisp $\mathrm{N}$ model of $NF$. Our proof does not depend on Holmes' Tangled Type Theory ($\mathit{TTT}$). It establishes that if $\mathit{ZF}$ is consistent, then $\mathit{NF}$ is also consistent. It achieves that via the chain $\mathit{ZF} \rightarrow$ Fuzzy $\mathit{ZF} \rightarrow Σ\rightarrow \mathit{NF}$. The method presented in this paper offers a novel perspective connecting fuzzy logic with classic set theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14400 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the consistency of NF via Fuzzy Forcing Simón, Nicolás Sevilla Logic In this paper, we present a proof of the consistency of the New Foundations set theory ($\mathit{NF}$). $\mathit{NF}$'s main idea is to permit very large sets (including the Universal Set) by restricting set formation to stratified formulas, thereby avoiding the classic set-theoretic paradoxes. Our proof employs a new forcing method incorporating concepts from fuzzy logic. A brief outline of the proof can be as follows: (1) We extend $ZF$ to Fuzzy $\mathit{ZF}$ with a membership function $μ$ over $D=\mathbb{Q} \cap [0,1]$; (2) we define Fuzzy $\mathit{NF}$ as $Σ$, and (3) we derive a crisp $\mathrm{N}$ model of $NF$. Our proof does not depend on Holmes' Tangled Type Theory ($\mathit{TTT}$). It establishes that if $\mathit{ZF}$ is consistent, then $\mathit{NF}$ is also consistent. It achieves that via the chain $\mathit{ZF} \rightarrow$ Fuzzy $\mathit{ZF} \rightarrow Σ\rightarrow \mathit{NF}$. The method presented in this paper offers a novel perspective connecting fuzzy logic with classic set theory. |
| title | On the consistency of NF via Fuzzy Forcing |
| topic | Logic |
| url | https://arxiv.org/abs/2504.14400 |