The initial segment condition for $κ^+$-supercompactness
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918053611044864 |
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| author | Schlutzenberg, Farmer |
| author_facet | Schlutzenberg, Farmer |
| contents | We give a development of the fine structure of mice with long extenders, to the level of $κ^+$-supercompact cardinals $κ$. We do this using a hierarchy with features more analogous to those familiar in the short extender context than the hierarchies introduced by Woodin and by Neeman-Steel. In particular, the mice we consider satisfy stronger versions of the initial segment condition. We establish a form of fine structural condensation involving embeddings $π:H\to M$ which need not be the identity below the projectum of $H$ (under special assumptions). We also adapt the analysis of the Dodd structure of short extenders on the sequence to mice at this level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13827 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The initial segment condition for $κ^+$-supercompactness Schlutzenberg, Farmer Logic 03E45, 03E55 We give a development of the fine structure of mice with long extenders, to the level of $κ^+$-supercompact cardinals $κ$. We do this using a hierarchy with features more analogous to those familiar in the short extender context than the hierarchies introduced by Woodin and by Neeman-Steel. In particular, the mice we consider satisfy stronger versions of the initial segment condition. We establish a form of fine structural condensation involving embeddings $π:H\to M$ which need not be the identity below the projectum of $H$ (under special assumptions). We also adapt the analysis of the Dodd structure of short extenders on the sequence to mice at this level. |
| title | The initial segment condition for $κ^+$-supercompactness |
| topic | Logic 03E45, 03E55 |
| url | https://arxiv.org/abs/2306.13827 |