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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2509.14222 |
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| _version_ | 1866916955073544192 |
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| author | Murwanashyaka, Juvenal |
| author_facet | Murwanashyaka, Juvenal |
| contents | Harvey Friedman's $ \mathsf{WD} $ is a weak set theory given by the following non-logical axioms: $ \mathsf{(W)} \; \forall x y \, \exists z \, \forall u \left[ \, u \in z \leftrightarrow ( \, u \in x \; \vee \; u = y \, ) \, \right] $; $ \mathsf{(D)} \; \forall x y \, \exists z \, \forall u \, \left[ \, u \in z \leftrightarrow ( \, u \in x \; \wedge \; u \neq y \, ) \, \right] $. We answer a question raised by Albert Visser which asks whether $ \mathsf{WD} $ is parameter-free sequential. Let $ \mathsf{WD} + \mathsf{EXT} $ denote the theory we obtain by extending $ \mathsf{WD} $ with the axiom of extensionality. We show that $ \mathsf{WD} + \mathsf{EXT} $, and hence also $ \mathsf{WD} $, is not parameter-free sequential by using forcing to construct a model $ \mathcal{V}^{ \star } $ of $ \mathsf{WD} + \mathsf{EXT} $ where $ \left( \mathcal{V}^{ \star } , a \right) \simeq \left( \mathcal{V}^{ \star } , b \right) $ for any two elements $a, b $ of $ \mathcal{V}^{ \star } $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14222 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Friedman's $ \mathsf{WD} $ is not parameter-free sequential Murwanashyaka, Juvenal Logic Harvey Friedman's $ \mathsf{WD} $ is a weak set theory given by the following non-logical axioms: $ \mathsf{(W)} \; \forall x y \, \exists z \, \forall u \left[ \, u \in z \leftrightarrow ( \, u \in x \; \vee \; u = y \, ) \, \right] $; $ \mathsf{(D)} \; \forall x y \, \exists z \, \forall u \, \left[ \, u \in z \leftrightarrow ( \, u \in x \; \wedge \; u \neq y \, ) \, \right] $. We answer a question raised by Albert Visser which asks whether $ \mathsf{WD} $ is parameter-free sequential. Let $ \mathsf{WD} + \mathsf{EXT} $ denote the theory we obtain by extending $ \mathsf{WD} $ with the axiom of extensionality. We show that $ \mathsf{WD} + \mathsf{EXT} $, and hence also $ \mathsf{WD} $, is not parameter-free sequential by using forcing to construct a model $ \mathcal{V}^{ \star } $ of $ \mathsf{WD} + \mathsf{EXT} $ where $ \left( \mathcal{V}^{ \star } , a \right) \simeq \left( \mathcal{V}^{ \star } , b \right) $ for any two elements $a, b $ of $ \mathcal{V}^{ \star } $. |
| title | Friedman's $ \mathsf{WD} $ is not parameter-free sequential |
| topic | Logic |
| url | https://arxiv.org/abs/2509.14222 |