Good projective witnesses
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866913810702401536 |
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| author | Fischer, Vera Friedman, Sy David Schrittesser, David Törnquist, Asger |
| author_facet | Fischer, Vera Friedman, Sy David Schrittesser, David Törnquist, Asger |
| contents | We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality $\mathfrak a_{\text{g}}$ of a maximal cofinitary group (MCG) is strictly between $\aleph_1$ and $\mathfrak{c}$, and there is a $Π^1_2$-definable MCG of this cardinality. Here $Π^1_2$ is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1904_05823 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Good projective witnesses Fischer, Vera Friedman, Sy David Schrittesser, David Törnquist, Asger Logic 03E17, 03E35 We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality $\mathfrak a_{\text{g}}$ of a maximal cofinitary group (MCG) is strictly between $\aleph_1$ and $\mathfrak{c}$, and there is a $Π^1_2$-definable MCG of this cardinality. Here $Π^1_2$ is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum. |
| title | Good projective witnesses |
| topic | Logic 03E17, 03E35 |
| url | https://arxiv.org/abs/1904.05823 |