Proper classes of maximal $θ$-independent families from large cardinals
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866911994469154816 |
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| author | Ryan-Smith, Calliope |
| author_facet | Ryan-Smith, Calliope |
| contents | While maximal independent families can be constructed from ZFC via Zorn's lemma, the presence of a maximal $σ$-independent family already gives an inner model with a measurable cardinal, and Kunen has shown that from a measurable cardinal one can construct a forcing extension in which there is a maximal $σ$-independent family. We extend this technique to construct proper classes of maximal $θ$-independent families for various uncountable $θ$. In the first instance, a single $θ^+$-strongly compact cardinal has a set-generic extension with a proper class of maximal $θ$-independent families. In the second, we take a class-generic extension of a model with a proper class of measurable cardinals to obtain a proper class of $θ$ for which there is a maximal $θ$-independent family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10137 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Proper classes of maximal $θ$-independent families from large cardinals Ryan-Smith, Calliope Logic 03E35 (Primary) 03E55, 03E05 (Secondary) While maximal independent families can be constructed from ZFC via Zorn's lemma, the presence of a maximal $σ$-independent family already gives an inner model with a measurable cardinal, and Kunen has shown that from a measurable cardinal one can construct a forcing extension in which there is a maximal $σ$-independent family. We extend this technique to construct proper classes of maximal $θ$-independent families for various uncountable $θ$. In the first instance, a single $θ^+$-strongly compact cardinal has a set-generic extension with a proper class of maximal $θ$-independent families. In the second, we take a class-generic extension of a model with a proper class of measurable cardinals to obtain a proper class of $θ$ for which there is a maximal $θ$-independent family. |
| title | Proper classes of maximal $θ$-independent families from large cardinals |
| topic | Logic 03E35 (Primary) 03E55, 03E05 (Secondary) |
| url | https://arxiv.org/abs/2408.10137 |