Intermediate models and Kinna--Wagner Principles
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908715170398208 |
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| author | Karagila, Asaf Schilhan, Jonathan |
| author_facet | Karagila, Asaf Schilhan, Jonathan |
| contents | Kinna--Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write $\mathsf{KWP}$ to denote that there is some $α$ for which this holds. The Kinna--Wagner Conjecture, formulated by the first author in [9], states that if $V$ is a model of $\mathsf{ZF}+\mathsf{KWP}$ and $G$ is a $V$-generic filter, then whenever $W$ is an intermediate model of $\mathsf{ZF}$, that is $V\subseteq W\subseteq V[G]$, then $W=V(x)$ for some $x$ if and only if $W$ satisfies $\mathsf{KWP}$. In this work we prove the conjecture and generalise it even further. We include a brief historical overview of Kinna--Wagner Principles and new results about Kinna--Wagner Principles in the multiverse of sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intermediate models and Kinna--Wagner Principles Karagila, Asaf Schilhan, Jonathan Logic Primary 03E25, Secondary 03E35, 03E40 Kinna--Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write $\mathsf{KWP}$ to denote that there is some $α$ for which this holds. The Kinna--Wagner Conjecture, formulated by the first author in [9], states that if $V$ is a model of $\mathsf{ZF}+\mathsf{KWP}$ and $G$ is a $V$-generic filter, then whenever $W$ is an intermediate model of $\mathsf{ZF}$, that is $V\subseteq W\subseteq V[G]$, then $W=V(x)$ for some $x$ if and only if $W$ satisfies $\mathsf{KWP}$. In this work we prove the conjecture and generalise it even further. We include a brief historical overview of Kinna--Wagner Principles and new results about Kinna--Wagner Principles in the multiverse of sets. |
| title | Intermediate models and Kinna--Wagner Principles |
| topic | Logic Primary 03E25, Secondary 03E35, 03E40 |
| url | https://arxiv.org/abs/2409.07352 |